Taylor series - Knowledge

8796: 7281: 8791:{\displaystyle {\begin{aligned}\sin x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)!}}x^{2n+1}&&=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-\cdots &&{\text{for all }}x\\\cos x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n)!}}x^{2n}&&=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots &&{\text{for all }}x\\\tan x&=\sum _{n=1}^{\infty }{\frac {B_{2n}(-4)^{n}\left(1-4^{n}\right)}{(2n)!}}x^{2n-1}&&=x+{\frac {x^{3}}{3}}+{\frac {2x^{5}}{15}}+\cdots &&{\text{for }}|x|<{\frac {\pi }{2}}\\\sec x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}E_{2n}}{(2n)!}}x^{2n}&&=1+{\frac {x^{2}}{2}}+{\frac {5x^{4}}{24}}+\cdots &&{\text{for }}|x|<{\frac {\pi }{2}}\\\arcsin x&=\sum _{n=0}^{\infty }{\frac {(2n)!}{4^{n}(n!)^{2}(2n+1)}}x^{2n+1}&&=x+{\frac {x^{3}}{6}}+{\frac {3x^{5}}{40}}+\cdots &&{\text{for }}|x|\leq 1\\\arccos x&={\frac {\pi }{2}}-\arcsin x\\&={\frac {\pi }{2}}-\sum _{n=0}^{\infty }{\frac {(2n)!}{4^{n}(n!)^{2}(2n+1)}}x^{2n+1}&&={\frac {\pi }{2}}-x-{\frac {x^{3}}{6}}-{\frac {3x^{5}}{40}}-\cdots &&{\text{for }}|x|\leq 1\\\arctan x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{2n+1}}x^{2n+1}&&=x-{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}-\cdots &&{\text{for }}|x|\leq 1,\ x\neq \pm i\end{aligned}}} 9868: 8863: 16078: 9863:{\displaystyle {\begin{aligned}\sinh x&=\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{(2n+1)!}}&&=x+{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}+\cdots &&{\text{for all }}x\\\cosh x&=\sum _{n=0}^{\infty }{\frac {x^{2n}}{(2n)!}}&&=1+{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}+\cdots &&{\text{for all }}x\\\tanh x&=\sum _{n=1}^{\infty }{\frac {B_{2n}4^{n}\left(4^{n}-1\right)}{(2n)!}}x^{2n-1}&&=x-{\frac {x^{3}}{3}}+{\frac {2x^{5}}{15}}-{\frac {17x^{7}}{315}}+\cdots &&{\text{for }}|x|<{\frac {\pi }{2}}\\\operatorname {arsinh} x&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n)!}{4^{n}(n!)^{2}(2n+1)}}x^{2n+1}&&=x-{\frac {x^{3}}{6}}+{\frac {3x^{5}}{40}}-\cdots &&{\text{for }}|x|\leq 1\\\operatorname {artanh} x&=\sum _{n=0}^{\infty }{\frac {x^{2n+1}}{2n+1}}&&=x+{\frac {x^{3}}{3}}+{\frac {x^{5}}{5}}+\cdots &&{\text{for }}|x|\leq 1,\ x\neq \pm 1\end{aligned}}} 15043: 16073:{\displaystyle {\begin{aligned}T(x_{1},\ldots ,x_{d})&=\sum _{n_{1}=0}^{\infty }\cdots \sum _{n_{d}=0}^{\infty }{\frac {(x_{1}-a_{1})^{n_{1}}\cdots (x_{d}-a_{d})^{n_{d}}}{n_{1}!\cdots n_{d}!}}\,\left({\frac {\partial ^{n_{1}+\cdots +n_{d}}f}{\partial x_{1}^{n_{1}}\cdots \partial x_{d}^{n_{d}}}}\right)(a_{1},\ldots ,a_{d})\\&=f(a_{1},\ldots ,a_{d})+\sum _{j=1}^{d}{\frac {\partial f(a_{1},\ldots ,a_{d})}{\partial x_{j}}}(x_{j}-a_{j})+{\frac {1}{2!}}\sum _{j=1}^{d}\sum _{k=1}^{d}{\frac {\partial ^{2}f(a_{1},\ldots ,a_{d})}{\partial x_{j}\partial x_{k}}}(x_{j}-a_{j})(x_{k}-a_{k})\\&\qquad \qquad +{\frac {1}{3!}}\sum _{j=1}^{d}\sum _{k=1}^{d}\sum _{l=1}^{d}{\frac {\partial ^{3}f(a_{1},\ldots ,a_{d})}{\partial x_{j}\partial x_{k}\partial x_{l}}}(x_{j}-a_{j})(x_{k}-a_{k})(x_{l}-a_{l})+\cdots \end{aligned}}} 14975: 7252: 14346: 6677: 18519: 14970:{\displaystyle {\begin{aligned}(1+x)e^{x}&=e^{x}+xe^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}+\sum _{n=0}^{\infty }{\frac {x^{n+1}}{n!}}=1+\sum _{n=1}^{\infty }{\frac {x^{n}}{n!}}+\sum _{n=0}^{\infty }{\frac {x^{n+1}}{n!}}\\&=1+\sum _{n=1}^{\infty }{\frac {x^{n}}{n!}}+\sum _{n=1}^{\infty }{\frac {x^{n}}{(n-1)!}}=1+\sum _{n=1}^{\infty }\left({\frac {1}{n!}}+{\frac {1}{(n-1)!}}\right)x^{n}\\&=1+\sum _{n=1}^{\infty }{\frac {n+1}{n!}}x^{n}\\&=\sum _{n=0}^{\infty }{\frac {n+1}{n!}}x^{n}.\end{aligned}}} 19720: 13554: 3437: 7247:{\displaystyle {\begin{aligned}(1+x)^{\frac {1}{2}}&=1+{\frac {1}{2}}x-{\frac {1}{8}}x^{2}+{\frac {1}{16}}x^{3}-{\frac {5}{128}}x^{4}+{\frac {7}{256}}x^{5}-\cdots &=\sum _{n=0}^{\infty }{\frac {(-1)^{n-1}(2n)!}{4^{n}(n!)^{2}(2n-1)}}x^{n},\\(1+x)^{-{\frac {1}{2}}}&=1-{\frac {1}{2}}x+{\frac {3}{8}}x^{2}-{\frac {5}{16}}x^{3}+{\frac {35}{128}}x^{4}-{\frac {63}{256}}x^{5}+\cdots &=\sum _{n=0}^{\infty }{\frac {(-1)^{n}(2n)!}{4^{n}(n!)^{2}}}x^{n}.\end{aligned}}} 16899: 20029: 2817: 13111: 12627: 3869: 3830: 33: 5882: 3063:, and told that Newton had developed a general method for expanding functions in series. Newton had in fact used a cumbersome method involving long division of series and term-by-term integration, but Gregory did not know it and set out to discover a general method for himself. In early 1671 Gregory discovered something like the general Maclaurin series and sent a letter to Collins including series for 2486: 3838: 12278: 16683: 6248: 10924: 13549:{\displaystyle {\begin{aligned}e^{x}&=\left(c_{0}+c_{1}x+c_{2}x^{2}+c_{3}x^{3}+c_{4}x^{4}+\cdots \right)\left(1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots \right)\\&=c_{0}+c_{1}x+\left(c_{2}-{\frac {c_{0}}{2}}\right)x^{2}+\left(c_{3}-{\frac {c_{1}}{2}}\right)x^{3}+\left(c_{4}-{\frac {c_{2}}{2}}+{\frac {c_{0}}{4!}}\right)x^{4}+\cdots \end{aligned}}} 10601: 17322: 13897: 5585: 17951: 1484: 18234: 10322: 10085: 1800: 16484: 2812:{\displaystyle {\begin{aligned}\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}&={\frac {x^{0}}{0!}}+{\frac {x^{1}}{1!}}+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+{\frac {x^{5}}{5!}}+\cdots \\&=1+x+{\frac {x^{2}}{2}}+{\frac {x^{3}}{6}}+{\frac {x^{4}}{24}}+{\frac {x^{5}}{120}}+\cdots .\end{aligned}}} 5981: 16882: 11126: 10626: 11726:

Several methods exist for the calculation of Taylor series of a large number of functions. One can attempt to use the definition of the Taylor series, though this often requires generalizing the form of the coefficients according to a readily apparent pattern. Alternatively, one can use manipulations

3397:(the Gudermannian function). However, thinking that he had merely redeveloped a method by Newton, Gregory never described how he obtained these series, and it can only be inferred that he understood the general method by examining scratch work he had scribbled on the back of another letter from 1671. 11717: 11422: 10337: 18505:

Finally, in practice one wants to approximate the function with a finite number of terms, say with a Taylor polynomial or a partial sum of the trigonometric series, respectively. In the case of the Taylor series the error is very small in a neighbourhood of the point where it is computed, while it

17585: 17020: 13630: 16457: 17594: 12622:{\displaystyle {\begin{aligned}f(x)&=\ln {\bigl (}1+(\cos x-1){\bigr )}\\&=(\cos x-1)-{\tfrac {1}{2}}(\cos x-1)^{2}+{\tfrac {1}{3}}(\cos x-1)^{3}+O{\left((\cos x-1)^{4}\right)}\\&=-{\frac {x^{2}}{2}}-{\frac {x^{4}}{12}}-{\frac {x^{6}}{45}}+O{\left(x^{8}\right)}.\end{aligned}}\!} 14337: 5239: 17960: 6555: 10098: 9909: 4826: 11727:

such as substitution, multiplication or division, addition or subtraction of standard Taylor series to construct the Taylor series of a function, by virtue of Taylor series being power series. In some cases, one can also derive the Taylor series by repeatedly applying

5877:{\displaystyle {\begin{aligned}\ln(1-x)&=-\sum _{n=1}^{\infty }{\frac {x^{n}}{n}}=-x-{\frac {x^{2}}{2}}-{\frac {x^{3}}{3}}-\cdots ,\\\ln(1+x)&=\sum _{n=1}^{\infty }(-1)^{n+1}{\frac {x^{n}}{n}}=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-\cdots .\end{aligned}}} 4616: 16746: 5442: 10937: 1205: 12268: 4267: 2337: 11855: 2468: 15011:" in diverse areas of mathematics. In particular, this is true in areas where the classical definitions of functions break down. For example, using Taylor series, one may extend analytic functions to sets of matrices and operators, such as the 11431: 4995: 14138: 11139: 17331: 1569: 4018: 7286: 12831: 18485:

The convergence of both series has very different properties. Even if the Taylor series has positive convergence radius, the resulting series may not coincide with the function; but if the function is analytic then the series converges

16142: 13076: 16678:{\displaystyle T(\mathbf {x} )=f(\mathbf {a} )+(\mathbf {x} -\mathbf {a} )^{\mathsf {T}}Df(\mathbf {a} )+{\frac {1}{2!}}(\mathbf {x} -\mathbf {a} )^{\mathsf {T}}\left\{D^{2}f(\mathbf {a} )\right\}(\mathbf {x} -\mathbf {a} )+\cdots ,} 12272:

The first several terms from the second series can be substituted into each term of the first series. Because the first term in the second series has degree 2, three terms of the first series suffice to give a 7th-degree polynomial:

2167: 6243:{\displaystyle {\begin{aligned}{\frac {1}{1-x}}&=\sum _{n=0}^{\infty }x^{n}\\{\frac {1}{(1-x)^{2}}}&=\sum _{n=1}^{\infty }nx^{n-1}\\{\frac {1}{(1-x)^{3}}}&=\sum _{n=2}^{\infty }{\frac {(n-1)n}{2}}x^{n-2}.\end{aligned}}} 5204: 12140: 18373: 3273: 10919:{\displaystyle {\begin{aligned}{\frac {2}{\pi }}K(x)&=\sum _{n=0}^{\infty }{\frac {^{2}}{16^{n}(n!)^{4}}}x^{2n}\\{\frac {2}{\pi }}E(x)&=\sum _{n=0}^{\infty }{\frac {^{2}}{(1-2n)16^{n}(n!)^{4}}}x^{2n}\end{aligned}}} 6399: 14173: 5561: 17025: 8868: 6411: 12926: 10596:{\displaystyle {\begin{aligned}{\text{Ti}}_{2}(x)&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{2}}}x^{2n+1}\\{\text{Ti}}_{3}(x)&=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{3}}}x^{2n+1}\end{aligned}}} 3341: 17317:{\displaystyle {\begin{aligned}f_{x}&=e^{x}\ln(1+y)\\f_{y}&={\frac {e^{x}}{1+y}}\\f_{xx}&=e^{x}\ln(1+y)\\f_{yy}&=-{\frac {e^{x}}{(1+y)^{2}}}\\f_{xy}&=f_{yx}={\frac {e^{x}}{1+y}}.\end{aligned}}} 4673: 13892:{\displaystyle c_{0}=1,\ \ c_{1}=1,\ \ c_{2}-{\tfrac {1}{2}}c_{0}={\tfrac {1}{2}},\ \ c_{3}-{\tfrac {1}{2}}c_{1}={\tfrac {1}{6}},\ \ c_{4}-{\tfrac {1}{2}}c_{2}+{\tfrac {1}{24}}c_{0}={\tfrac {1}{24}},\ \ldots .} 17946:{\displaystyle {\begin{aligned}T(x,y)=&f(a,b)+(x-a)f_{x}(a,b)+(y-b)f_{y}(a,b)\\&{}+{\frac {1}{2!}}\left((x-a)^{2}f_{xx}(a,b)+2(x-a)(y-b)f_{xy}(a,b)+(y-b)^{2}f_{yy}(a,b)\right)+\cdots \end{aligned}}} 13116: 4471: 11940: 5303: 3574: 195: 2040: 17965: 17599: 17336: 15048: 14351: 12283: 12149: 10942: 10631: 10342: 10103: 9914: 6682: 5986: 5590: 2491: 4158: 19348: 2190: 12704: 3395: 18229:{\displaystyle {\begin{aligned}T(x,y)&=0+0(x-0)+1(y-0)+{\frac {1}{2}}{\big (}0(x-0)^{2}+2(x-0)(y-0)+(-1)(y-0)^{2}{\big )}+\cdots \\&=y+xy-{\tfrac {1}{2}}y^{2}+\cdots \end{aligned}}} 11748: 3665: 10317:{\displaystyle {\begin{aligned}\chi _{2}(x)&=\sum _{n=0}^{\infty }{\frac {1}{(2n+1)^{2}}}x^{2n+1}\\\chi _{3}(x)&=\sum _{n=0}^{\infty }{\frac {1}{(2n+1)^{3}}}x^{2n+1}\end{aligned}}} 2357: 4835: 14017: 10080:{\displaystyle {\begin{aligned}{\text{Li}}_{2}(x)&=\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}x^{n}\\{\text{Li}}_{3}(x)&=\sum _{n=1}^{\infty }{\frac {1}{n^{3}}}x^{n}\end{aligned}}} 3756:. The Taylor series can be used to calculate the value of an entire function at every point, if the value of the function, and of all of its derivatives, are known at a single point. 1901: 3910: 17011: 12713: 18502:, but additional requirements are needed to ensure the pointwise or uniform convergence (for instance, if the function is periodic and of class C then the convergence is uniform). 18449:. In contrast, the Fourier series is computed by integrating over an entire interval, so there is generally no such point where all the finite truncations of the series are exact. 12935: 12024: 18855: 11986: 2065: 14147:

Here we employ a method called "indirect expansion" to expand the given function. This method uses the known Taylor expansion of the exponential function. In order to expand

16877:{\displaystyle T(\mathbf {x} )=\sum _{|\alpha |\geq 0}{\frac {(\mathbf {x} -\mathbf {a} )^{\alpha }}{\alpha !}}\left({\mathrm {\partial } ^{\alpha }}f\right)(\mathbf {a} ),} 5073: 12035: 3814:

Approximations using the first few terms of a Taylor series can make otherwise unsolvable problems possible for a restricted domain; this approach is often used in physics.

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of a Taylor series can be zero. There are even infinitely differentiable functions defined on the real line whose Taylor series have a radius of convergence 0 everywhere.

11121:{\displaystyle {\begin{aligned}\vartheta _{00}(x)&=1+2\sum _{n=1}^{\infty }x^{n^{2}}\\\vartheta _{01}(x)&=1+2\sum _{n=1}^{\infty }(-1)^{n}x^{n^{2}}\end{aligned}}} 1479:{\displaystyle f(a)+{\frac {f'(a)}{1!}}(x-a)+{\frac {f''(a)}{2!}}(x-a)^{2}+{\frac {f'''(a)}{3!}}(x-a)^{3}+\cdots =\sum _{n=0}^{\infty }{\frac {f^{(n)}(a)}{n!}}(x-a)^{n}.} 13594: 3175: 6604: 6286: 5932: 18261: 3090: 3025: 16115: 14009: 13624: 13104: 6309: 13982: 13926: 5464: 5290: 3196: 13955: 3146: 3118: 3055: 2997: 2968: 19911: 12840: 11712:{\displaystyle \vartheta _{00}(x)^{1/6}\vartheta _{01}(x)^{-1/3}{\biggl }^{1/24}=\sum _{n=0}^{\infty }Q(n)x^{n}=\prod _{k=1}^{\infty }{\frac {1}{1-x^{2k-1}}}} 11417:{\displaystyle \vartheta _{00}(x)^{-1/6}\vartheta _{01}(x)^{-2/3}{\biggl }^{-1/24}=\sum _{n=0}^{\infty }P(n)x^{n}=\prod _{k=1}^{\infty }{\frac {1}{1-x^{k}}}} 17580:{\displaystyle {\begin{aligned}f_{x}(0,0)&=0\\f_{y}(0,0)&=1\\f_{xx}(0,0)&=0\\f_{yy}(0,0)&=-1\\f_{xy}(0,0)&=f_{yx}(0,0)=1.\end{aligned}}} 1795:{\displaystyle f(0)+{\frac {f'(0)}{1!}}x+{\frac {f''(0)}{2!}}x^{2}+{\frac {f'''(0)}{3!}}x^{3}+\cdots =\sum _{n=0}^{\infty }{\frac {f^{(n)}(0)}{n!}}x^{n}.} 4642:. The series is precisely the Taylor series, except that divided differences appear in place of differentiation: the series is formally similar to the 16452:{\displaystyle f(a,b)+(x-a)f_{x}(a,b)+(y-b)f_{y}(a,b)+{\frac {1}{2!}}{\Big (}(x-a)^{2}f_{xx}(a,b)+2(x-a)(y-b)f_{xy}(a,b)+(y-b)^{2}f_{yy}(a,b){\Big )}} 3202: 1070:

at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after

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The Taylor series is defined for a function which has infinitely many derivatives at a single point, whereas the Fourier series is defined for any

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is the function which is equal to its own derivative everywhere, and assumes the value 1 at the origin. However, one may equally well define an

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follows from Taylor series expansions for trigonometric and exponential functions. This result is of fundamental importance in such fields as

19901: 18408:). In this sense, the Fourier series is analogous to Taylor series, since the latter allows one to express a function as an infinite sum of 3289: 19994: 1132:

of the Taylor polynomials. A function may differ from the sum of its Taylor series, even if its Taylor series is convergent. A function is

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It was not until 1715 that a general method for constructing these series for all functions for which they exist was finally published by

1948: 291: 14332:{\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+\cdots .} 2905: 3833:

The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full period centered at the origin.

12648: 3767:) of the series can be used as approximations of the function. These approximations are good if sufficiently many terms are included. 3400:

In 1691–1692, Isaac Newton wrote down an explicit statement of the Taylor and Maclaurin series in an unpublished version of his work

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In the 14th century, the earliest examples of specific Taylor series (but not the general method) were given by Indian mathematician

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considered the problem of summing an infinite series to achieve a finite result, but rejected it as an impossibility; the result was

6550:{\displaystyle {\binom {\alpha }{n}}=\prod _{k=1}^{n}{\frac {\alpha -k+1}{k}}={\frac {\alpha (\alpha -1)\cdots (\alpha -n+1)}{n!}}.} 3601: 3404:. However, this work was never completed and the relevant sections were omitted from the portions published in 1704 under the title 19835: 18506:

may be very large at a distant point. In the case of the Fourier series the error is distributed along the domain of the function.

4821:{\displaystyle \sum _{n=0}^{\infty }{\frac {u^{n}}{n!}}\Delta ^{n}a_{i}=e^{-u}\sum _{j=0}^{\infty }{\frac {u^{j}}{j!}}a_{i+j}.} 1845: 1120:

gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is

19845: 19435: 19300: 19281: 19227: 18997: 18978: 18627: 4654:, the terms in the series converge to the terms of the Taylor series, and in this sense generalizes the usual Taylor series. 4116: 2881:

proposed a philosophical resolution of the paradox, but the mathematical content was apparently unresolved until taken up by

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Hofmann, Josef Ehrenfried (1939). "On the Discovery of the Logarithmic Series and Its Development in England up to Cotes".

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and so the power series expansion agrees with the Taylor series. Thus a function is analytic in an open disk centered at

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A second-order Taylor series expansion of a scalar-valued function of more than one variable can be written compactly as

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The (truncated) series can be used to compute function values numerically, (often by recasting the polynomial into the

1183: 1033: 596: 4611:{\displaystyle \lim _{h\to 0^{+}}\sum _{n=0}^{\infty }{\frac {t^{n}}{n!}}{\frac {\Delta _{h}^{n}f(a)}{h^{n}}}=f(a+t).} 114: 19989: 19469: 19414: 19250: 19148: 19117: 19024: 18891: 18875: 18839: 18823: 18762: 4374:, which might have singularities, never converge to a value different from the function itself. The complex function 2928:). During the following two centuries his followers developed further series expansions and rational approximations. 552: 270: 18611: 4410:; in these cases, one can often still achieve a series expansion if one allows also negative powers of the variable 3346: 19999: 5437:{\displaystyle e^{x}=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+\cdots .} 537: 17: 19891: 19881: 3276: 873: 547: 522: 204: 1112:

of the function. Taylor polynomials are approximations of a function, which become generally more accurate as

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Differentiation and integration of power series can be performed term by term and is hence particularly easy.

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of a point, whereas the computation of the Fourier series requires knowing the function on its whole domain

12263:{\displaystyle \cos x-1=-{\frac {x^{2}}{2}}+{\frac {x^{4}}{24}}-{\frac {x^{6}}{720}}+O{\left(x^{8}\right)}.} 20032: 5226:

Several important Maclaurin series expansions follow. All these expansions are valid for complex arguments

4323: 2936: 309: 281: 6620:, this is essentially the infinite geometric series mentioned in the previous section. The special cases 4262:{\displaystyle f(x)={\begin{cases}e^{-1/x^{2}}&{\text{if }}x\neq 0\\0&{\text{if }}x=0\end{cases}}} 392: 20014: 19501: 18453: 5221: 4395:

in the Taylor series of an infinitely differentiable function defined on the real line, a consequence of

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sine and cosine, are examples of entire functions. Examples of functions that are not entire include the

3422:, a Scottish mathematician, who published the special case of the Taylor result in the mid-18th century. 2332:{\displaystyle (x-1)-{\tfrac {1}{2}}(x-1)^{2}+{\tfrac {1}{3}}(x-1)^{3}-{\tfrac {1}{4}}(x-1)^{4}+\cdots ,} 906: 514: 352: 324: 18460:. In a certain sense one could say that the Taylor series is "local" and the Fourier series is "global". 11991: 20063: 19896: 18495: 18494:

on every compact subset of the convergence interval. Concerning the Fourier series, if the function is

11947: 11850:{\displaystyle f(x)=\ln(\cos x),\quad x\in {\bigl (}{-{\tfrac {\pi }{2}}},{\tfrac {\pi }{2}}{\bigr )},} 777: 741: 518: 397: 286: 276: 2463:{\displaystyle \ln a+{\frac {1}{a}}(x-a)-{\frac {1}{a^{2}}}{\frac {\left(x-a\right)^{2}}{2}}+\cdots .} 20058: 19886: 19876: 19866: 16474: 4990:{\displaystyle f(a+t)=\lim _{h\to 0^{+}}e^{-t/h}\sum _{j=0}^{\infty }f(a+jh){\frac {(t/h)^{j}}{j!}}.} 4407: 4272: 2867: 1121: 541: 14133:{\displaystyle {\frac {e^{x}}{\cos x}}=1+x+x^{2}+{\tfrac {2}{3}}x^{3}+{\tfrac {1}{2}}x^{4}+\cdots .} 377: 19496: 10328: 8858:

have Maclaurin series closely related to the series for the corresponding trigonometric functions:

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if and only if its Taylor series converges to the value of the function at each point of the disk.

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that an infinite number of progressive subdivisions could be performed to achieve a finite result.

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As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows

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version of the first equation of this paragraph, with a full analogy to the single variable case.

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The generalization of the Taylor series does converge to the value of the function itself for any

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Malet, Antoni (1993). "James Gregorie on Tangents and the "Taylor" Rule for Series Expansions".

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The computation of Taylor series requires the knowledge of the function on an arbitrary small

14992:(including those discussed above) are defined by some property that holds for them, such as a 12826:{\displaystyle e^{x}=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+\cdots ,} 6573: 6255: 5901: 3150: 19850: 19595: 19425: 18954: 18904: 18885: 18869: 18865: 18849: 18833: 18803: 18772: 18487: 14993: 13071:{\displaystyle {\frac {e^{x}}{\cos x}}=c_{0}+c_{1}x+c_{2}x^{2}+c_{3}x^{3}+c_{4}x^{4}+\cdots } 10606: 10090: 4400: 3753: 3283: 1000: 666: 437: 382: 343: 249: 16085: 13988: 13599: 13083: 3066: 3001: 19961: 19798: 19567: 18554: 18532: 18479: 16738: 14997: 13960: 13904: 11728: 7257: 6404: 5268: 5263: 5243: 5209: 5023: 4371: 4363: 3803:

Algebraic operations can be done readily on the power series representation; for instance,

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In other areas, such as formal analysis, it is more convenient to work directly with the

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a power series which, one hopes to prove, is the Taylor series of the desired solution.

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describe the world of the elliptic modular functions and they have these Taylor series:

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is not the zero function, so does not equal its Taylor series around the origin. Thus,

19871: 19782: 19767: 19739: 19719: 19658: 19335: 19327: 19191: 19137: 19093: 19064: 18959: 18524: 16462: 15012: 14985: 5199:{\displaystyle f(a+t)=\lim _{h\to 0^{+}}\int _{-\infty }^{\infty }f(a+x)dP_{t/h,h}(x).} 4452: 3804: 3797: 1156:. This implies that the function is analytic at every point of the interval (or disk). 1145: 945: 848: 832: 772: 767: 762: 726: 607: 526: 432: 427: 231: 226: 15038:

The Taylor series may also be generalized to functions of more than one variable with

12135:{\displaystyle \ln(1+x)=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}+O{\left(x^{4}\right)}} 19971: 19772: 19744: 19698: 19688: 19668: 19653: 19515: 19465: 19431: 19410: 19339: 19296: 19277: 19246: 19239: 19223: 19162: 19144: 19123: 19113: 19097: 19068: 19020: 19013: 18993: 18974: 18945: 18537: 18518: 18393: 15001: 10618: 9889: 8821: 5891: 5571: 4123: 3808: 3774: 3764: 3725: 3475: 3431: 2874: 2057: 1133: 1129: 1019: 853: 631: 509: 462: 319: 314: 19203: 1082:, who made extensive use of this special case of Taylor series in the 18th century. 19956: 19777: 19703: 19693: 19673: 19575: 19371: 19367: 19319: 19269: 19215: 19183: 19085: 19056: 15016: 5973: 4441: 4396: 4367: 4135: 3786: 3737: 1838: 863: 757: 731: 592: 504: 468: 19518: 18412:. Nevertheless, the two series differ from each other in several relevant issues: 4131: 19734: 19663: 18968: 18955:

Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables

13957:

can thus be computed one at a time, amounting to long division of the series for

6301: 5887: 5576: 5294: 5026: 3692: 3419: 1079: 995: 868: 822: 817: 704: 617: 562: 19273: 19219: 18368:{\displaystyle e^{x}\ln(1+y)=y+xy-{\tfrac {1}{2}}y^{2}+\cdots ,\qquad |y|<1.} 3268:{\textstyle \ln \,\tan {\tfrac {1}{2}}{{\bigl (}{\tfrac {1}{2}}\pi +x{\bigr )}}} 19966: 19951: 19946: 19625: 19610: 19105: 18548: 18409: 18389: 18383: 16734: 12027: 10929: 6394:{\displaystyle (1+x)^{\alpha }=\sum _{n=0}^{\infty }{\binom {\alpha }{n}}x^{n}} 5000: 4415: 3793: 3782: 2925: 1556: 1191: 878: 686: 453: 19060: 18467:. In particular, the function could be nowhere differentiable. (For example, 5556:{\displaystyle \exp(\exp {x}-1)=\sum _{n=0}^{\infty }{\frac {B_{n}}{n!}}x^{n}} 20047: 19605: 19127: 12632: 9901: 6567: 4643: 4330: 1552: 1149: 1141: 858: 372: 329: 5459:

is the exponential function of the predecessor of the exponential function:

19936: 19678: 19620: 19402: 19169:. AMS Colloquium Publications. Vol. 31. American Mathematical Society. 19008: 18949: 15023: 8843: 4126:

at all. In fact, the set of functions with a convergent Taylor series is a

3412: 3060: 2870: 2044:

By integrating the above Maclaurin series, we find the Maclaurin series of

1200: 1071: 612: 357: 2904:. Though no record of his work survives, writings of his followers in the 19683: 19630: 19158: 17326:

Evaluating these derivatives at the origin gives the Taylor coefficients

16887: 12921:{\displaystyle \cos x=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-\cdots .} 11743:

In order to compute the 7th degree Maclaurin polynomial for the function

6665: 5456: 4456: 4392: 4370:

always possess a convergent Taylor series, and even the Taylor series of

4282:, and has all derivatives zero there. Consequently, the Taylor series of 3709: 3474:

in the complex plane (or an interval in the real line), it is said to be

3436: 1187: 1086: 1047: 975: 19331: 16929:

In order to compute a second-order Taylor series expansion around point

11426:

The strict partition number sequence Q(n) has that generating function:

4391:

More generally, every sequence of real or complex numbers can appear as

4142:

does converge, its limit need not be equal to the value of the function

19532: 19323: 19195: 19089: 4127: 4051:

In contrast, also shown is a picture of the natural logarithm function

3336:{\textstyle \operatorname {arcsec} {\bigl (}{\sqrt {2}}e^{x}{\bigr )},} 2921: 2886: 2882: 1810: 1517: 1097: 1067: 721: 645: 367: 362: 266: 16898: 15026:

themselves. Thus one may define a solution of a differential equation

19615: 19523: 3713: 2878: 1494: 650: 640: 19187: 19076:

Feigenbaum, L. (1985). "Brook Taylor and the method of increments".

18396:(or a function defined on a closed interval ) as an infinite sum of 5215: 19563: 18970:

The Calculus Lifesaver: All the Tools You Need to Excel at Calculus

16902:

Second-order Taylor series approximation (in orange) of a function

16702: 4066:. These approximations converge to the function only in the region 3864:, Taylor polynomials of higher degree provide worse approximations. 3455:: the Taylor series is identically 0, although the function is not. 716: 458: 415: 104: 18714: 16468: 4074:; outside of this region the higher-degree Taylor polynomials are 19047:

Dani, S. G. (2012). "Ancient Indian Mathematics – A Conspectus".

3470:

is given by a convergent power series in an open disk centred at

2894: 18587: 10621:

of first kind K and of second kind E can be defined as follows:

3868: 1078:

when 0 is the point where the derivatives are considered, after

1074:, who introduced them in 1715. A Taylor series is also called a 18896: 18894: 18405: 8801: 3721: 2917: 2897:

independently employed a similar method a few centuries later.

2885:, as it had been prior to Aristotle by the Presocratic Atomist 4388:

in the complex plane and its Taylor series is undefined at 0.

5238: 4406:

A function cannot be written as a Taylor series centred at a

3829: 32: 19208:"1. Test Functions §1.1. A review of Differential Calculus" 18401: 4255: 2913: 19110:

An introduction to probability theory and its applications

17015:

one first computes all the necessary partial derivatives:

11935:{\displaystyle f(x)={\ln }{\bigl (}1+(\cos x-1){\bigr )},} 3759:

Uses of the Taylor series for analytic functions include:

19397:. Vol. 1 (2nd ed.). Cambridge University Press. 3837: 3569:{\displaystyle f(x)=\sum _{n=0}^{\infty }a_{n}(x-b)^{n}.} 7276:

and their inverses have the following Maclaurin series:

4115:. Taylor's theorem can be used to obtain a bound on the 18783: 18781: 12643:

Suppose we want the Taylor series at 0 of the function

2341:

and more generally, the corresponding Taylor series of

1140:

if it is equal to the sum of its Taylor series in some

1066:

of terms that are expressed in terms of the function's

43:

and its Taylor approximations by polynomials of degree

19513: 18315: 18195: 16886:

which is to be understood as a still more abbreviated

14100: 14075: 13866: 13841: 13816: 13782: 13757: 13723: 13698: 12430: 12387: 12026:

The Taylor series for the natural logarithm is (using

11826: 11810: 3818: 3373: 3349: 3292: 3237: 3217: 3205: 3153: 3125: 3097: 3069: 3034: 3004: 2976: 2947: 2287: 2250: 2213: 2129: 2104: 2079: 19912:

1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)

19902:

1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)

19395:

Series and Products in the Development of Mathematics

18750: 18599: 18264: 17963: 17597: 17334: 17023: 16947: 16749: 16487: 16145: 16088: 15046: 15033: 14349: 14176: 14020: 13991: 13963: 13934: 13907: 13633: 13602: 13564: 13114: 13086: 12938: 12843: 12716: 12651: 12281: 12152: 12038: 11994: 11950: 11867: 11751: 11434: 11142: 10940: 10629: 10340: 10101: 9912: 8866: 7284: 6680: 6576: 6414: 6312: 6258: 5984: 5904: 5588: 5467: 5306: 5271: 5076: 4838: 4676: 4474: 4161: 4040:) the error is less than 0.08215. In particular, for 3913: 3604: 3494: 3184: 2858:

in the denominator of each term in the infinite sum.

2489: 2360: 2193: 2068: 1951: 1848: 1572: 1208: 117: 18988:

Bilodeau, Gerald; Thie, Paul; Keough, G. E. (2010).

18778: 18695:, p. 21–23, see Prop. VII, Thm. 3, Cor. 2. See 18514: 17589:

Substituting these values in to the general formula

16125:, the Taylor series to second order about the point 12635:, the coefficients for all the odd powers are zero. 4384:

approaches 0 along the imaginary axis, so it is not

2821:

The above expansion holds because the derivative of

2035:{\displaystyle 1-(x-1)+(x-1)^{2}-(x-1)^{3}+\cdots .} 19036:"Methodus Incrementorum Directa & Inversa]" 18811: 19238: 19136: 19012: 18987: 18790: 18367: 18228: 17945: 17579: 17316: 17005: 16876: 16677: 16451: 16109: 16072: 14969: 14331: 14132: 14003: 13976: 13949: 13920: 13891: 13618: 13588: 13548: 13098: 13070: 12920: 12825: 12708:The Taylor series for the exponential function is 12698: 12621: 12262: 12134: 12018: 11980: 11934: 11849: 11711: 11416: 11120: 10918: 10595: 10316: 10079: 9862: 8790: 7246: 6598: 6549: 6393: 6280: 6242: 5926: 5876: 5555: 5436: 5284: 5198: 4989: 4820: 4610: 4261: 4012: 3905:. The pink curve is a polynomial of degree seven: 3849:only provide accurate approximations in the range 3659: 3568: 3389: 3335: 3267: 3190: 3169: 3140: 3112: 3084: 3049: 3019: 2991: 2962: 2811: 2462: 2331: 2161: 2034: 1895: 1794: 1478: 189: 19241:Mathematical Thought from Ancient to Modern Times 18377: 16444: 16270: 12695: 12618: 11586: 11508: 11297: 11219: 6431: 6418: 6375: 6362: 5216:List of Maclaurin series of some common functions 4009: 3687:is equal to the sum of its Taylor series for all 20045: 18944: 18900: 18881: 18861: 18845: 18829: 18799: 18768: 18699:, pp. 329–332 for English translation, and 16741:the Taylor series for several variables becomes 15007:Taylor series are used to define functions and " 5099: 4861: 4476: 4451:, and this can be done by using the calculus of 4022:The error in this approximation is no more than 2908:suggest that he found the Taylor series for the 190:{\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)} 19460:Thomas, George B. Jr.; Finney, Ross L. (1996). 19259: 18925: 18643: 18416:The finite truncations of the Taylor series of 16469:Second-order Taylor series in several variables 14979: 12699:{\displaystyle g(x)={\frac {e^{x}}{\cos x}}.\!} 11721: 3724:. For these functions the Taylor series do not 3390:{\textstyle 2\arctan e^{x}-{\tfrac {1}{2}}\pi } 19212:The analysis of partial differential operators 4455:. Specifically, the following theorem, due to 3660:{\displaystyle {\frac {f^{(n)}(b)}{n!}}=a_{n}} 19548: 19295:(2nd ed.). University of Chicago Press. 19157: 19006: 18741: 18633: 18158: 18053: 14162:, we use the known Taylor series of function 12347: 12313: 11924: 11890: 11839: 11800: 5894:(since it was published in his 1668 treatise 4668:, the following power series identity holds: 4638:th finite difference operator with step size 3325: 3301: 3259: 3231: 1027: 19995:Hypergeometric function of a matrix argument 19459: 19453:Direct and Reverse Methods of Incrementation 19260:Kolk, Johan A.C.; Duistermaat, J.J. (2010). 18593: 18551: – Power series with rational exponents 16475:Linearization § Multivariable functions 10326:And the formulas presented below are called 9895: 4085:incurred in approximating a function by its 19851:1 + 1/2 + 1/3 + ... (Riemann zeta function) 19481:James Gregory; Tercentenary Memorial Volume 18545:– best approximation by a rational function 16461:where the subscripts denote the respective 4032:. For a full cycle centered at the origin ( 19555: 19541: 19075: 18706: 14988:are defined by an algebraic equation, and 13080:Multiplying both sides by the denominator 7267: 5976:and its derivatives have Maclaurin series 4138:. Even if the Taylor series of a function 4089:th-degree Taylor polynomial is called the 4059:and some of its Taylor polynomials around 2906:Kerala school of astronomy and mathematics 1034: 1020: 19907:1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series) 19448:Methodus Incrementorum Directa et Inversa 19349:"The Discovery of the Series Formula for 19202: 19134: 18919: 15281: 13106:and then expanding it as a series yields 11576: 11287: 5258:terms of its Taylor series at 0 (in red). 3891:Pictured is an accurate approximation of 3209: 3157: 150: 19562: 19478: 19290: 19214:. Vol. 1 (2nd ed.). Springer. 18664: 18605: 16897: 12930:Assume the series for their quotient is 11731:. Particularly convenient is the use of 10609:these formulas are of great importance. 5237: 4362:even if they converge. By contrast, the 4048:, the error is less than 0.000003. 3867: 3836: 3828: 3435: 3415:, after whom the series are now named. 1896:{\displaystyle 1+x+x^{2}+x^{3}+\cdots .} 31: 27:Mathematical approximation of a function 19479:Turnbull, Herbert Westren, ed. (1939). 19262:"Taylor Expansion in Several Variables" 19173: 18817: 8849: 6403:whose coefficients are the generalized 5233: 1566:, the Maclaurin series takes the form: 558:Differentiating under the integral sign 14: 20046: 19444: 19427:A Source Book in Mathematics 1200–1800 19423: 19266:Distributions: Theory and applications 19104: 18966: 18756: 18735: 18696: 18692: 18577: 16607: 16544: 11859:one may first rewrite the function as 4657:In general, for any infinite sequence 4122:In general, Taylor series need not be 3486:is given by a convergent power series 1555:. This series can be written by using 19536: 19514: 19401: 19312:Archive for History of Exact Sciences 19309: 19245:. New York: Oxford University Press. 19236: 19112:. Vol. 2 (3rd ed.). Wiley. 19078:Archive for History of Exact Sciences 19033: 19019:(2nd ed.). John Wiley and Sons. 18720: 18700: 18675: 18617: 17006:{\displaystyle f(x,y)=e^{x}\ln(1+y),} 10612: 5898:). Both of these series converge for 3425: 3418:The Maclaurin series was named after 2843:equals 1. This leaves the terms 1559:, as in the right side formula. With 19046: 18649: 7260:is retained, this simplifies to the 6570:and has value 1.) It converges for 6289: 5565: 5251:(in blue), and the sum of the first 4432:can be written as a Laurent series. 4380:, however, does not approach 0 when 4333:, this example shows that there are 19872:1 − 1 + 1 − 1 + ⋯ (Grandi's series) 19392: 19353:by Leibniz, Gregory and Nilakantha" 19346: 19167:Functional analysis and semi-groups 18670: 11134:P(n) has this generating function: 5967: 4335:infinitely differentiable functions 3819:Approximation error and convergence 3691:in the complex plane, it is called 2171:The corresponding Taylor series of 24: 18992:. Jones & Bartlett Publisher. 16841: 15957: 15944: 15931: 15882: 15714: 15701: 15652: 15543: 15500: 15358: 15330: 15290: 15150: 15119: 15034:Taylor series in several variables 14922: 14858: 14756: 14692: 14646: 14581: 14535: 14477: 14431: 14206: 12019:{\displaystyle x\mapsto \cos x-1.} 11670: 11624: 11384: 11338: 11073: 10996: 10811: 10679: 10513: 10389: 10252: 10148: 10041: 9961: 9712: 9474: 9221: 9068: 8903: 8619: 8387: 8123: 7918: 7681: 7507: 7321: 7151: 6857: 6422: 6366: 6354: 6295: 6185: 6103: 6030: 5768: 5640: 5516: 5336: 5134: 5129: 4920: 4772: 4721: 4693: 4545: 4514: 3526: 2510: 1735: 1528:. The derivative of order zero of 1407: 99:Part of a series of articles about 25: 20075: 19990:Generalized hypergeometric series 19489: 19293:The Beginnings of Western Science 12638: 11981:{\displaystyle x\mapsto \ln(1+x)} 11944:the composition of two functions 6288:. These are special cases of the 5212:implies that the identity holds. 4435: 4078:approximations for the function. 20028: 20027: 20000:Lauricella hypergeometric series 19718: 19464:(9th ed.). Addison Wesley. 19139:Advanced Engineering Mathematics 18791:Bilodeau, Thie & Keough 2010 18517: 16864: 16808: 16800: 16757: 16659: 16651: 16635: 16597: 16589: 16560: 16534: 16526: 16512: 16495: 14142: 11738: 9904:have these defining identities: 3406:Tractatus de Quadratura Curvarum 20010:Riemann's differential equation 19143:(2nd ed.). Prentice Hall. 18910: 18726: 18723:, p. 418, See Exercise 13. 18345: 16117:that depends on two variables, 15796: 15795: 11791: 8813:appearing in the expansions of 6606:for any real or complex number 5934:. (In addition, the series for 4999:The series on the right is the 4099:and is denoted by the function 19462:Calculus and Analytic Geometry 19372:10.1080/0025570X.1990.11977541 18973:. Princeton University Press. 18683: 18655: 18378:Comparison with Fourier series 18355: 18347: 18293: 18281: 18147: 18134: 18131: 18122: 18116: 18104: 18101: 18089: 18074: 18061: 18035: 18023: 18014: 18002: 17983: 17971: 17925: 17913: 17891: 17878: 17872: 17860: 17844: 17832: 17829: 17817: 17808: 17796: 17774: 17761: 17726: 17714: 17701: 17689: 17683: 17671: 17658: 17646: 17640: 17628: 17617: 17605: 17564: 17552: 17529: 17517: 17484: 17472: 17442: 17430: 17400: 17388: 17361: 17349: 17230: 17217: 17174: 17162: 17073: 17061: 16997: 16985: 16963: 16951: 16868: 16860: 16813: 16796: 16781: 16773: 16761: 16753: 16663: 16647: 16639: 16631: 16602: 16585: 16564: 16556: 16539: 16522: 16516: 16508: 16499: 16491: 16439: 16427: 16405: 16392: 16386: 16374: 16358: 16346: 16343: 16331: 16322: 16310: 16288: 16275: 16247: 16235: 16222: 16210: 16204: 16192: 16179: 16167: 16161: 16149: 16104: 16092: 16057: 16031: 16028: 16002: 15999: 15973: 15926: 15894: 15785: 15759: 15756: 15730: 15696: 15664: 15585: 15559: 15538: 15506: 15470: 15438: 15422: 15390: 15231: 15204: 15185: 15158: 15086: 15054: 14802: 14790: 14722: 14710: 14366: 14354: 13944: 13938: 13574: 13568: 13558:Comparing the coefficients of 12661: 12655: 12500: 12481: 12460: 12441: 12417: 12398: 12380: 12362: 12342: 12324: 12295: 12289: 12057: 12045: 11998: 11975: 11963: 11954: 11919: 11901: 11877: 11871: 11785: 11773: 11761: 11755: 11638: 11632: 11562: 11555: 11533: 11526: 11486: 11479: 11452: 11445: 11352: 11346: 11273: 11266: 11244: 11237: 11197: 11190: 11160: 11153: 11088: 11078: 11038: 11032: 10961: 10955: 10887: 10877: 10864: 10849: 10838: 10831: 10822: 10819: 10785: 10779: 10737: 10727: 10706: 10699: 10690: 10687: 10653: 10647: 10558: 10542: 10531: 10521: 10487: 10481: 10434: 10418: 10407: 10397: 10363: 10357: 10279: 10263: 10226: 10220: 10175: 10159: 10122: 10116: 10015: 10009: 9935: 9929: 9828: 9820: 9666: 9658: 9562: 9547: 9538: 9528: 9510: 9501: 9492: 9482: 9421: 9413: 9289: 9280: 9098: 9089: 8945: 8930: 8756: 8748: 8637: 8627: 8573: 8565: 8456: 8441: 8432: 8422: 8404: 8395: 8296: 8288: 8192: 8177: 8168: 8158: 8140: 8131: 8070: 8062: 7969: 7960: 7936: 7926: 7865: 7857: 7758: 7749: 7712: 7702: 7545: 7536: 7525: 7515: 7365: 7350: 7339: 7329: 7215: 7205: 7187: 7178: 7169: 7159: 6987: 6974: 6951: 6936: 6927: 6917: 6899: 6890: 6875: 6865: 6698: 6685: 6586: 6578: 6530: 6512: 6506: 6494: 6326: 6313: 6268: 6260: 6205: 6193: 6150: 6137: 6068: 6055: 5914: 5906: 5783: 5773: 5742: 5730: 5611: 5599: 5494: 5474: 5190: 5184: 5154: 5142: 5106: 5092: 5080: 4964: 4949: 4943: 4928: 4868: 4854: 4842: 4602: 4590: 4568: 4562: 4483: 4300:is identically zero. However, 4171: 4165: 3872:The Taylor approximations for 3785:. This makes the machinery of 3630: 3624: 3619: 3613: 3554: 3541: 3504: 3498: 2889:. It was through Archimedes's 2395: 2383: 2348:at an arbitrary nonzero point 2311: 2298: 2274: 2261: 2237: 2224: 2206: 2194: 2014: 2001: 1989: 1976: 1970: 1958: 1765: 1759: 1754: 1748: 1686: 1680: 1642: 1636: 1605: 1599: 1582: 1576: 1464: 1451: 1437: 1431: 1426: 1420: 1373: 1360: 1346: 1340: 1317: 1304: 1290: 1284: 1267: 1255: 1241: 1235: 1218: 1212: 1096:terms of a Taylor series is a 184: 178: 169: 163: 147: 141: 13: 1: 20005:Modular hypergeometric series 19846:1/4 + 1/16 + 1/64 + 1/256 + ⋯ 19176:National Mathematics Magazine 18937: 18498:then the series converges in 12835:and the series for cosine is 3887:, the approximations diverge. 3736:. That is, the Taylor series 1553:are both defined to be 1 1159: 484:Integral of inverse functions 19430:. Harvard University Press. 18901:Abramowitz & Stegun 1970 18882:Abramowitz & Stegun 1970 18862:Abramowitz & Stegun 1970 18846:Abramowitz & Stegun 1970 18830:Abramowitz & Stegun 1970 18800:Abramowitz & Stegun 1970 18769:Abramowitz & Stegun 1970 16082:For example, for a function 14980:Taylor series as definitions 12144:and for the cosine function 11735:to calculate Taylor series. 11722:Calculation of Taylor series 9881:appearing in the series for 8800:All angles are expressed in 5886:The last series is known as 4324:non-analytic smooth function 4153:. For example, the function 2472:The Maclaurin series of the 7: 20015:Theta hypergeometric series 19502:Encyclopedia of Mathematics 19274:10.1007/978-0-8176-4675-2_6 19220:10.1007/978-3-642-61497-2_2 19135:Greenberg, Michael (1998). 18990:An Introduction to Analysis 18926:Kolk & Duistermaat 2010 18922:, See Eqq. 1.1.7 and 1.1.7′ 18510: 6292:given in the next section. 5222:List of mathematical series 3841:The Taylor polynomials for 3796:and evaluating it with the 2935:was shown in a letter from 1804: 907:Calculus on Euclidean space 325:Logarithmic differentiation 10: 20080: 19897:Infinite arithmetic series 19841:1/2 + 1/4 + 1/8 + 1/16 + ⋯ 19836:1/2 − 1/4 + 1/8 − 1/16 + ⋯ 19409:. New Delhi: McGraw-Hill. 18381: 16893: 16472: 13589:{\displaystyle g(x)\cos x} 10607:statistical thermodynamics 5219: 3822: 3777:is uniquely extended to a 3720:tangent, and its inverse, 3478:in this region. Thus for 3429: 2861: 1813:is the polynomial itself. 20023: 19980: 19924: 19859: 19828: 19821: 19791: 19760: 19753: 19727: 19716: 19639: 19583: 19574: 19455:] (in Latin). London. 19407:Real and Complex Analysis 19061:10.1007/s12045-012-0022-y 18742:Hille & Phillips 1957 18634:Boyer & Merzbach 1991 18437:are all exactly equal to 18392:enables one to express a 13596:with the coefficients of 11132:partition number sequence 10329:inverse tangent integrals 9896:Polylogarithmic functions 4273:infinitely differentiable 3170:{\textstyle \ln \,\sec x} 2939:several Maclaurin series 2868:ancient Greek philosopher 1809:The Taylor series of any 1184:infinitely differentiable 641:Summand limit (term test) 19393:—— (2021) . 19291:Lindberg, David (2007). 19015:A History of Mathematics 18594:Thomas & Finney 1996 18565: 14990:transcendental functions 11733:computer algebra systems 10093:are defined as follows: 6599:{\displaystyle |x|<1} 6281:{\displaystyle |x|<1} 5927:{\displaystyle |x|<1} 4347:whose Taylor series are 3744:if the distance between 1816:The Maclaurin series of 320:Implicit differentiation 310:Differentiation notation 237:Inverse function theorem 19728:Properties of sequences 18967:Banner, Adrian (2007). 18738:, p. 230–232 18398:trigonometric functions 12631:Since the cosine is an 7274:trigonometric functions 7268:Trigonometric functions 6252:All are convergent for 5580:) has Maclaurin series 5298:) has Maclaurin series 4030:| / 9! 3781:on an open disk in the 3706:trigonometric functions 3085:{\textstyle \arctan x,} 3020:{\textstyle \arcsin x,} 2910:trigonometric functions 2902:Madhava of Sangamagrama 1916:, the Taylor series of 1524:evaluated at the point 1170:complex-valued function 1164:The Taylor series of a 783:Helmholtz decomposition 19591:Arithmetic progression 19445:Taylor, Brook (1715). 19424:Struik, D. J. (1969). 18560:Function approximation 18369: 18230: 17947: 17581: 17318: 17007: 16926: 16878: 16679: 16453: 16111: 16110:{\displaystyle f(x,y)} 16074: 15877: 15856: 15835: 15647: 15626: 15496: 15154: 15123: 15004:by its Taylor series. 14971: 14926: 14862: 14760: 14696: 14650: 14585: 14539: 14481: 14435: 14333: 14210: 14158:as a Taylor series in 14134: 14005: 14004:{\displaystyle \cos x} 13978: 13951: 13922: 13893: 13620: 13619:{\displaystyle e^{x},} 13590: 13550: 13100: 13099:{\displaystyle \cos x} 13072: 12922: 12827: 12700: 12623: 12264: 12136: 12020: 11982: 11936: 11851: 11713: 11674: 11628: 11418: 11388: 11342: 11122: 11077: 11000: 10930:Jacobi theta functions 10920: 10815: 10683: 10597: 10517: 10393: 10318: 10256: 10152: 10091:Legendre chi functions 10081: 10045: 9965: 9864: 9716: 9478: 9225: 9072: 8907: 8792: 8623: 8391: 8127: 7922: 7685: 7511: 7325: 7262:binomial approximation 7248: 7155: 6861: 6600: 6551: 6460: 6395: 6358: 6282: 6244: 6189: 6107: 6034: 5928: 5878: 5772: 5644: 5557: 5520: 5438: 5340: 5286: 5259: 5200: 4991: 4924: 4822: 4776: 4697: 4612: 4518: 4263: 4014: 3888: 3865: 3834: 3763:The partial sums (the 3718:trigonometric function 3661: 3570: 3530: 3456: 3402:De Quadratura Curvarum 3391: 3337: 3269: 3192: 3171: 3142: 3114: 3086: 3051: 3021: 2993: 2964: 2813: 2514: 2464: 2333: 2163: 2036: 1897: 1796: 1739: 1480: 1411: 917:Limit of distributions 737:Directional derivative 393:Faà di Bruno's formula 191: 93: 19982:Hypergeometric series 19596:Geometric progression 19483:. G. Bell & Sons. 18490:to the function, and 18370: 18231: 17948: 17582: 17319: 17008: 16901: 16879: 16680: 16454: 16112: 16075: 15857: 15836: 15815: 15627: 15606: 15476: 15127: 15096: 14994:differential equation 14972: 14906: 14842: 14740: 14676: 14630: 14565: 14519: 14461: 14415: 14334: 14190: 14135: 14006: 13979: 13977:{\displaystyle e^{x}} 13952: 13923: 13921:{\displaystyle c_{i}} 13894: 13621: 13591: 13551: 13101: 13073: 12923: 12828: 12701: 12624: 12265: 12137: 12021: 11983: 11937: 11852: 11714: 11654: 11608: 11419: 11368: 11322: 11123: 11057: 10980: 10921: 10795: 10663: 10598: 10497: 10373: 10319: 10236: 10132: 10082: 10025: 9945: 9865: 9696: 9458: 9205: 9052: 8887: 8793: 8603: 8371: 8107: 7902: 7665: 7491: 7305: 7249: 7135: 6841: 6601: 6566:, this product is an 6552: 6440: 6405:binomial coefficients 6396: 6338: 6283: 6245: 6169: 6087: 6014: 5949:, and the series for 5929: 5879: 5752: 5624: 5558: 5500: 5444:It converges for all 5439: 5320: 5287: 5285:{\displaystyle e^{x}} 5241: 5201: 5029:that takes the value 4992: 4904: 4823: 4756: 4677: 4613: 4498: 4401:radius of convergence 4372:meromorphic functions 4364:holomorphic functions 4264: 4117:size of the remainder 4015: 3871: 3840: 3832: 3754:radius of convergence 3662: 3571: 3510: 3439: 3392: 3338: 3284:Gudermannian function 3270: 3193: 3191:{\displaystyle \tan } 3172: 3143: 3115: 3087: 3052: 3022: 2994: 2965: 2851:in the numerator and 2814: 2494: 2465: 2334: 2164: 2037: 1898: 1797: 1719: 1481: 1391: 1001:Mathematical analysis 912:Generalized functions 597:arithmetico-geometric 438:Leibniz integral rule 192: 35: 19962:Trigonometric series 19754:Properties of series 19601:Harmonic progression 19360:Mathematics Magazine 19347:Roy, Ranjan (1990). 18555:Approximation theory 18533:Asymptotic expansion 18480:Weierstrass function 18262: 17961: 17595: 17332: 17021: 16945: 16747: 16739:multi-index notation 16485: 16143: 16086: 15044: 14998:exponential function 14347: 14174: 14018: 13989: 13961: 13950:{\displaystyle g(x)} 13932: 13905: 13631: 13600: 13562: 13112: 13084: 12936: 12841: 12714: 12649: 12279: 12150: 12036: 11992: 11948: 11865: 11749: 11729:integration by parts 11432: 11140: 10938: 10627: 10338: 10099: 9910: 8864: 8856:hyperbolic functions 8850:Hyperbolic functions 8835:in the expansion of 7282: 6678: 6574: 6412: 6310: 6304:is the power series 6256: 5982: 5902: 5586: 5465: 5304: 5269: 5264:exponential function 5244:exponential function 5234:Exponential function 5210:law of large numbers 5074: 4836: 4674: 4646:. When the function 4472: 4399:. As a result, the 4159: 3911: 3779:holomorphic function 3697:exponential function 3602: 3586:times, then setting 3492: 3347: 3290: 3203: 3182: 3151: 3141:{\textstyle \sec x,} 3123: 3113:{\textstyle \tan x,} 3095: 3067: 3050:{\textstyle x\cot x} 3032: 3002: 2992:{\textstyle \cos x,} 2974: 2963:{\textstyle \sin x,} 2945: 2891:method of exhaustion 2487: 2474:exponential function 2358: 2191: 2066: 1949: 1905:So, by substituting 1846: 1570: 1206: 1089:formed by the first 1006:Nonstandard analysis 474:Lebesgue integration 344:Rules and identities 115: 19942:Formal power series 19034:Bruce, Ian (2007). 18703:for re-translation. 18465:integrable function 16463:partial derivatives 15382: 15354: 14996:. For example, the 14986:algebraic functions 5453:generating function 5138: 5024:Poisson-distributed 4558: 4445:continuous function 4322:is an example of a 3752:is larger than the 3695:. The polynomials, 3578:Differentiating by 3448:is not analytic at 1104:that is called the 677:Cauchy condensation 479:Contour integration 205:Fundamental theorem 132: 19740:Monotonic function 19659:Fibonacci sequence 19516:Weisstein, Eric W. 19324:10.1007/BF00375656 19237:Kline, M. (1990). 19163:Phillips, Ralph S. 19090:10.1007/bf00329903 19040:17centurymaths.com 18960:Dover Publications 18946:Abramowitz, Milton 18744:, pp. 300–327 18667:, pp. 168–174 18525:Mathematics portal 18388:The trigonometric 18365: 18324: 18226: 18224: 18204: 17943: 17941: 17577: 17575: 17314: 17312: 17003: 16927: 16925:around the origin. 16874: 16792: 16675: 16449: 16107: 16070: 16068: 15361: 15333: 15013:matrix exponential 14967: 14965: 14329: 14130: 14109: 14084: 14001: 13974: 13947: 13928:of the series for 13918: 13889: 13875: 13850: 13825: 13791: 13766: 13732: 13707: 13616: 13586: 13546: 13544: 13096: 13068: 12918: 12823: 12696: 12619: 12616: 12439: 12396: 12260: 12132: 12016: 11978: 11932: 11847: 11835: 11819: 11709: 11414: 11118: 11116: 10916: 10914: 10619:elliptic integrals 10613:Elliptic functions 10593: 10591: 10314: 10312: 10077: 10075: 9860: 9858: 8788: 8786: 7244: 7242: 6596: 6547: 6391: 6278: 6240: 6238: 5924: 5874: 5872: 5553: 5434: 5282: 5260: 5196: 5121: 5120: 4987: 4882: 4830:So in particular, 4818: 4608: 4544: 4497: 4453:finite differences 4259: 4254: 4010: 3889: 3866: 3835: 3798:Clenshaw algorithm 3765:Taylor polynomials 3657: 3582:the above formula 3566: 3457: 3446: 3426:Analytic functions 3387: 3382: 3333: 3265: 3246: 3226: 3188: 3167: 3138: 3110: 3082: 3047: 3017: 2989: 2960: 2809: 2807: 2460: 2329: 2296: 2259: 2222: 2159: 2138: 2113: 2088: 2032: 1893: 1792: 1476: 849:Partial derivative 778:generalized Stokes 672:Alternating series 553:Reduction formulae 542:Heaviside's method 523:tangent half-angle 510:Cylindrical shells 433:Integral transform 428:Lists of integrals 232:Mean value theorem 187: 118: 94: 20064:Series expansions 20041: 20040: 19972:Generating series 19920: 19919: 19892:1 − 2 + 4 − 8 + ⋯ 19887:1 + 2 + 4 + 8 + ⋯ 19882:1 − 2 + 3 − 4 + ⋯ 19877:1 + 2 + 3 + 4 + ⋯ 19867:1 + 1 + 1 + 1 + ⋯ 19817: 19816: 19745:Periodic sequence 19714: 19713: 19699:Triangular number 19689:Pentagonal number 19669:Heptagonal number 19654:Complete sequence 19576:Integer sequences 19437:978-0-674-82355-6 19302:978-0-226-48205-7 19283:978-0-8176-4672-1 19229:978-3-642-61497-2 18999:978-0-7637-7492-9 18980:978-0-691-13088-0 18962:. Ninth printing. 18538:Newton polynomial 18496:square-integrable 18394:periodic function 18323: 18203: 18049: 17754: 17305: 17240: 17120: 16831: 16767: 16583: 16266: 15971: 15813: 15728: 15604: 15557: 15384: 15279: 15002:analytic function 14948: 14884: 14809: 14779: 14729: 14671: 14612: 14560: 14508: 14456: 14318: 14293: 14268: 14231: 14108: 14083: 14044: 13901:The coefficients 13882: 13874: 13849: 13824: 13801: 13798: 13790: 13765: 13742: 13739: 13731: 13706: 13683: 13680: 13658: 13655: 13519: 13494: 13441: 13388: 13303: 13278: 12962: 12907: 12882: 12812: 12787: 12762: 12690: 12583: 12563: 12543: 12438: 12395: 12229: 12209: 12189: 12104: 12084: 11834: 11818: 11707: 11581: 11412: 11292: 10897: 10774: 10747: 10642: 10568: 10473: 10444: 10349: 10289: 10185: 10061: 10001: 9981: 9921: 9890:Bernoulli numbers 9843: 9817: 9803: 9783: 9752: 9655: 9641: 9616: 9566: 9436: 9410: 9396: 9371: 9346: 9296: 9180: 9166: 9141: 9105: 9027: 9013: 8988: 8952: 8822:Bernoulli numbers 8771: 8745: 8731: 8711: 8661: 8562: 8548: 8523: 8497: 8460: 8366: 8334: 8285: 8271: 8246: 8196: 8085: 8059: 8045: 8020: 7976: 7880: 7854: 7840: 7815: 7765: 7640: 7626: 7601: 7552: 7466: 7452: 7427: 7372: 7225: 7112: 7089: 7066: 7043: 7027: 7002: 6955: 6818: 6795: 6772: 6749: 6733: 6709: 6668:function and its 6542: 6483: 6429: 6373: 6215: 6160: 6078: 6005: 5896:Logarithmotechnia 5892:Nicholas Mercator 5859: 5839: 5813: 5709: 5689: 5660: 5572:natural logarithm 5566:Natural logarithm 5541: 5423: 5398: 5361: 5098: 5035:with probability 4982: 4860: 4797: 4718: 4582: 4539: 4475: 4241: 4218: 4004: 3979: 3954: 3898:around the point 3809:harmonic analysis 3775:analytic function 3642: 3442: 3432:Analytic function 3381: 3311: 3245: 3225: 3177:(the integral of 2794: 2774: 2754: 2734: 2689: 2664: 2639: 2614: 2589: 2564: 2535: 2449: 2416: 2381: 2295: 2258: 2221: 2137: 2112: 2087: 2058:natural logarithm 1777: 1698: 1654: 1617: 1532:is defined to be 1449: 1358: 1302: 1253: 1130:infinite sequence 1124:, its sum is the 1110:Taylor polynomial 1044: 1043: 924: 923: 886: 885: 854:Multiple integral 790: 789: 694: 693: 661:Direct comparison 632:Convergence tests 570: 569: 538:Partial fractions 405: 404: 315:Second derivative 16:(Redirected from 20071: 20059:Complex analysis 20031: 20030: 19957:Dirichlet series 19826: 19825: 19758: 19757: 19722: 19694:Polygonal number 19674:Hexagonal number 19647: 19581: 19580: 19557: 19550: 19543: 19534: 19533: 19529: 19528: 19510: 19484: 19475: 19456: 19441: 19420: 19398: 19389: 19387: 19386: 19380: 19374:. Archived from 19357: 19352: 19343: 19306: 19287: 19256: 19244: 19233: 19199: 19170: 19154: 19142: 19131: 19101: 19072: 19043: 19030: 19018: 19003: 18984: 18963: 18950:Stegun, Irene A. 18932: 18914: 18908: 18898: 18889: 18879: 18873: 18859: 18853: 18843: 18837: 18827: 18821: 18815: 18809: 18785: 18776: 18766: 18760: 18754: 18748: 18730: 18724: 18718: 18712: 18687: 18681: 18659: 18653: 18647: 18641: 18631: 18625: 18615: 18609: 18603: 18597: 18591: 18585: 18575: 18543:Padé approximant 18527: 18522: 18521: 18477: 18448: 18442: 18436: 18427:about the point 18426: 18374: 18372: 18371: 18366: 18358: 18350: 18335: 18334: 18325: 18316: 18274: 18273: 18255: 18253: 18245: 18235: 18233: 18232: 18227: 18225: 18215: 18214: 18205: 18196: 18172: 18162: 18161: 18155: 18154: 18082: 18081: 18057: 18056: 18050: 18042: 17952: 17950: 17949: 17944: 17942: 17932: 17928: 17912: 17911: 17899: 17898: 17859: 17858: 17795: 17794: 17782: 17781: 17755: 17753: 17742: 17737: 17732: 17713: 17712: 17670: 17669: 17586: 17584: 17583: 17578: 17576: 17551: 17550: 17516: 17515: 17471: 17470: 17429: 17428: 17387: 17386: 17348: 17347: 17323: 17321: 17320: 17315: 17313: 17306: 17304: 17293: 17292: 17283: 17278: 17277: 17258: 17257: 17241: 17239: 17238: 17237: 17215: 17214: 17205: 17193: 17192: 17155: 17154: 17138: 17137: 17121: 17119: 17108: 17107: 17098: 17089: 17088: 17054: 17053: 17037: 17036: 17012: 17010: 17009: 17004: 16978: 16977: 16941:of the function 16940: 16924: 16883: 16881: 16880: 16875: 16867: 16859: 16855: 16851: 16850: 16849: 16844: 16832: 16830: 16822: 16821: 16820: 16811: 16803: 16794: 16791: 16784: 16776: 16760: 16732: 16718: 16708: 16700: 16684: 16682: 16681: 16676: 16662: 16654: 16646: 16642: 16638: 16627: 16626: 16612: 16611: 16610: 16600: 16592: 16584: 16582: 16571: 16563: 16549: 16548: 16547: 16537: 16529: 16515: 16498: 16458: 16456: 16455: 16450: 16448: 16447: 16426: 16425: 16413: 16412: 16373: 16372: 16309: 16308: 16296: 16295: 16274: 16273: 16267: 16265: 16254: 16234: 16233: 16191: 16190: 16136: 16124: 16120: 16116: 16114: 16113: 16108: 16079: 16077: 16076: 16071: 16069: 16056: 16055: 16043: 16042: 16027: 16026: 16014: 16013: 15998: 15997: 15985: 15984: 15972: 15970: 15969: 15968: 15956: 15955: 15943: 15942: 15929: 15925: 15924: 15906: 15905: 15890: 15889: 15879: 15876: 15871: 15855: 15850: 15834: 15829: 15814: 15812: 15801: 15791: 15784: 15783: 15771: 15770: 15755: 15754: 15742: 15741: 15729: 15727: 15726: 15725: 15713: 15712: 15699: 15695: 15694: 15676: 15675: 15660: 15659: 15649: 15646: 15641: 15625: 15620: 15605: 15603: 15592: 15584: 15583: 15571: 15570: 15558: 15556: 15555: 15554: 15541: 15537: 15536: 15518: 15517: 15498: 15495: 15490: 15469: 15468: 15450: 15449: 15428: 15421: 15420: 15402: 15401: 15389: 15385: 15383: 15381: 15380: 15379: 15369: 15353: 15352: 15351: 15341: 15328: 15324: 15323: 15322: 15321: 15303: 15302: 15287: 15280: 15278: 15274: 15273: 15258: 15257: 15247: 15246: 15245: 15244: 15243: 15229: 15228: 15216: 15215: 15200: 15199: 15198: 15197: 15183: 15182: 15170: 15169: 15156: 15153: 15148: 15141: 15140: 15122: 15117: 15110: 15109: 15085: 15084: 15066: 15065: 15017:matrix logarithm 14976: 14974: 14973: 14968: 14966: 14959: 14958: 14949: 14947: 14939: 14928: 14925: 14920: 14899: 14895: 14894: 14885: 14883: 14875: 14864: 14861: 14856: 14829: 14825: 14824: 14815: 14811: 14810: 14808: 14785: 14780: 14778: 14767: 14759: 14754: 14730: 14728: 14708: 14707: 14698: 14695: 14690: 14672: 14670: 14662: 14661: 14652: 14649: 14644: 14617: 14613: 14611: 14603: 14602: 14587: 14584: 14579: 14561: 14559: 14551: 14550: 14541: 14538: 14533: 14509: 14507: 14499: 14498: 14483: 14480: 14475: 14457: 14455: 14447: 14446: 14437: 14434: 14429: 14411: 14410: 14395: 14394: 14378: 14377: 14338: 14336: 14335: 14330: 14319: 14317: 14309: 14308: 14299: 14294: 14292: 14284: 14283: 14274: 14269: 14267: 14259: 14258: 14249: 14232: 14230: 14222: 14221: 14212: 14209: 14204: 14186: 14185: 14167: 14161: 14157: 14139: 14137: 14136: 14131: 14120: 14119: 14110: 14101: 14095: 14094: 14085: 14076: 14070: 14069: 14045: 14043: 14032: 14031: 14022: 14012: 14010: 14008: 14007: 14002: 13983: 13981: 13980: 13975: 13973: 13972: 13956: 13954: 13953: 13948: 13927: 13925: 13924: 13919: 13917: 13916: 13898: 13896: 13895: 13890: 13880: 13876: 13867: 13861: 13860: 13851: 13842: 13836: 13835: 13826: 13817: 13811: 13810: 13799: 13796: 13792: 13783: 13777: 13776: 13767: 13758: 13752: 13751: 13740: 13737: 13733: 13724: 13718: 13717: 13708: 13699: 13693: 13692: 13681: 13678: 13668: 13667: 13656: 13653: 13643: 13642: 13625: 13623: 13622: 13617: 13612: 13611: 13595: 13593: 13592: 13587: 13555: 13553: 13552: 13547: 13545: 13535: 13534: 13525: 13521: 13520: 13518: 13510: 13509: 13500: 13495: 13490: 13489: 13480: 13475: 13474: 13457: 13456: 13447: 13443: 13442: 13437: 13436: 13427: 13422: 13421: 13404: 13403: 13394: 13390: 13389: 13384: 13383: 13374: 13369: 13368: 13348: 13347: 13335: 13334: 13319: 13315: 13311: 13304: 13302: 13294: 13293: 13284: 13279: 13277: 13269: 13268: 13259: 13246: 13242: 13235: 13234: 13225: 13224: 13212: 13211: 13202: 13201: 13189: 13188: 13179: 13178: 13163: 13162: 13150: 13149: 13128: 13127: 13105: 13103: 13102: 13097: 13077: 13075: 13074: 13069: 13061: 13060: 13051: 13050: 13038: 13037: 13028: 13027: 13015: 13014: 13005: 13004: 12989: 12988: 12976: 12975: 12963: 12961: 12950: 12949: 12940: 12927: 12925: 12924: 12919: 12908: 12906: 12898: 12897: 12888: 12883: 12881: 12873: 12872: 12863: 12832: 12830: 12829: 12824: 12813: 12811: 12803: 12802: 12793: 12788: 12786: 12778: 12777: 12768: 12763: 12761: 12753: 12752: 12743: 12726: 12725: 12705: 12703: 12702: 12697: 12691: 12689: 12678: 12677: 12668: 12628: 12626: 12625: 12620: 12617: 12610: 12609: 12605: 12604: 12584: 12579: 12578: 12569: 12564: 12559: 12558: 12549: 12544: 12539: 12538: 12529: 12518: 12514: 12513: 12509: 12508: 12507: 12468: 12467: 12440: 12431: 12425: 12424: 12397: 12388: 12355: 12351: 12350: 12317: 12316: 12269: 12267: 12266: 12261: 12256: 12255: 12251: 12250: 12230: 12225: 12224: 12215: 12210: 12205: 12204: 12195: 12190: 12185: 12184: 12175: 12141: 12139: 12138: 12133: 12131: 12130: 12126: 12125: 12105: 12100: 12099: 12090: 12085: 12080: 12079: 12070: 12025: 12023: 12022: 12017: 11987: 11985: 11984: 11979: 11941: 11939: 11938: 11933: 11928: 11927: 11894: 11893: 11887: 11856: 11854: 11853: 11848: 11843: 11842: 11836: 11827: 11821: 11820: 11811: 11804: 11803: 11718: 11716: 11715: 11710: 11708: 11706: 11705: 11704: 11676: 11673: 11668: 11650: 11649: 11627: 11622: 11604: 11603: 11599: 11590: 11589: 11582: 11580: 11571: 11570: 11569: 11554: 11553: 11541: 11540: 11525: 11524: 11514: 11512: 11511: 11505: 11504: 11500: 11478: 11477: 11468: 11467: 11463: 11444: 11443: 11423: 11421: 11420: 11415: 11413: 11411: 11410: 11409: 11390: 11387: 11382: 11364: 11363: 11341: 11336: 11318: 11317: 11313: 11301: 11300: 11293: 11291: 11282: 11281: 11280: 11265: 11264: 11252: 11251: 11236: 11235: 11225: 11223: 11222: 11216: 11215: 11211: 11189: 11188: 11179: 11178: 11174: 11152: 11151: 11127: 11125: 11124: 11119: 11117: 11113: 11112: 11111: 11110: 11096: 11095: 11076: 11071: 11031: 11030: 11017: 11016: 11015: 11014: 10999: 10994: 10954: 10953: 10925: 10923: 10922: 10917: 10915: 10911: 10910: 10898: 10896: 10895: 10894: 10876: 10875: 10847: 10846: 10845: 10817: 10814: 10809: 10775: 10767: 10761: 10760: 10748: 10746: 10745: 10744: 10726: 10725: 10715: 10714: 10713: 10685: 10682: 10677: 10643: 10635: 10602: 10600: 10599: 10594: 10592: 10588: 10587: 10569: 10567: 10566: 10565: 10540: 10539: 10538: 10519: 10516: 10511: 10480: 10479: 10474: 10471: 10464: 10463: 10445: 10443: 10442: 10441: 10416: 10415: 10414: 10395: 10392: 10387: 10356: 10355: 10350: 10347: 10323: 10321: 10320: 10315: 10313: 10309: 10308: 10290: 10288: 10287: 10286: 10258: 10255: 10250: 10219: 10218: 10205: 10204: 10186: 10184: 10183: 10182: 10154: 10151: 10146: 10115: 10114: 10086: 10084: 10083: 10078: 10076: 10072: 10071: 10062: 10060: 10059: 10047: 10044: 10039: 10008: 10007: 10002: 9999: 9992: 9991: 9982: 9980: 9979: 9967: 9964: 9959: 9928: 9927: 9922: 9919: 9887: 9880: 9869: 9867: 9866: 9861: 9859: 9841: 9831: 9823: 9818: 9815: 9812: 9804: 9799: 9798: 9789: 9784: 9779: 9778: 9769: 9755: 9753: 9751: 9737: 9736: 9718: 9715: 9710: 9669: 9661: 9656: 9653: 9650: 9642: 9637: 9636: 9635: 9622: 9617: 9612: 9611: 9602: 9588: 9586: 9585: 9567: 9565: 9546: 9545: 9527: 9526: 9516: 9500: 9499: 9480: 9477: 9472: 9437: 9429: 9424: 9416: 9411: 9408: 9405: 9397: 9392: 9391: 9390: 9377: 9372: 9367: 9366: 9365: 9352: 9347: 9342: 9341: 9332: 9318: 9316: 9315: 9297: 9295: 9278: 9277: 9273: 9266: 9265: 9251: 9250: 9241: 9240: 9227: 9224: 9219: 9181: 9178: 9175: 9167: 9165: 9157: 9156: 9147: 9142: 9140: 9132: 9131: 9122: 9108: 9106: 9104: 9087: 9086: 9074: 9071: 9066: 9028: 9025: 9022: 9014: 9012: 9004: 9003: 8994: 8989: 8987: 8979: 8978: 8969: 8955: 8953: 8951: 8928: 8927: 8909: 8906: 8901: 8841: 8834: 8819: 8812: 8797: 8795: 8794: 8789: 8787: 8769: 8759: 8751: 8746: 8743: 8740: 8732: 8727: 8726: 8717: 8712: 8707: 8706: 8697: 8683: 8681: 8680: 8662: 8660: 8646: 8645: 8644: 8625: 8622: 8617: 8576: 8568: 8563: 8560: 8557: 8549: 8544: 8543: 8542: 8529: 8524: 8519: 8518: 8509: 8498: 8490: 8482: 8480: 8479: 8461: 8459: 8440: 8439: 8421: 8420: 8410: 8393: 8390: 8385: 8367: 8359: 8351: 8335: 8327: 8299: 8291: 8286: 8283: 8280: 8272: 8267: 8266: 8265: 8252: 8247: 8242: 8241: 8232: 8218: 8216: 8215: 8197: 8195: 8176: 8175: 8157: 8156: 8146: 8129: 8126: 8121: 8086: 8078: 8073: 8065: 8060: 8057: 8054: 8046: 8041: 8040: 8039: 8026: 8021: 8016: 8015: 8006: 7992: 7990: 7989: 7977: 7975: 7958: 7957: 7956: 7944: 7943: 7924: 7921: 7916: 7881: 7873: 7868: 7860: 7855: 7852: 7849: 7841: 7836: 7835: 7834: 7821: 7816: 7811: 7810: 7801: 7787: 7785: 7784: 7766: 7764: 7747: 7746: 7742: 7741: 7740: 7720: 7719: 7701: 7700: 7687: 7684: 7679: 7641: 7638: 7635: 7627: 7625: 7617: 7616: 7607: 7602: 7600: 7592: 7591: 7582: 7568: 7566: 7565: 7553: 7551: 7534: 7533: 7532: 7513: 7510: 7505: 7467: 7464: 7461: 7453: 7451: 7443: 7442: 7433: 7428: 7426: 7418: 7417: 7408: 7394: 7392: 7391: 7373: 7371: 7348: 7347: 7346: 7327: 7324: 7319: 7253: 7251: 7250: 7245: 7243: 7236: 7235: 7226: 7224: 7223: 7222: 7204: 7203: 7193: 7177: 7176: 7157: 7154: 7149: 7123: 7122: 7113: 7105: 7100: 7099: 7090: 7082: 7077: 7076: 7067: 7059: 7054: 7053: 7044: 7036: 7028: 7020: 7005: 7004: 7003: 6995: 6966: 6965: 6956: 6954: 6935: 6934: 6916: 6915: 6905: 6889: 6888: 6863: 6860: 6855: 6829: 6828: 6819: 6811: 6806: 6805: 6796: 6788: 6783: 6782: 6773: 6765: 6760: 6759: 6750: 6742: 6734: 6726: 6711: 6710: 6702: 6663: 6662: 6660: 6659: 6656: 6653: 6641: 6640: 6638: 6637: 6634: 6631: 6619: 6609: 6605: 6603: 6602: 6597: 6589: 6581: 6565: 6556: 6554: 6553: 6548: 6543: 6541: 6533: 6489: 6484: 6479: 6462: 6459: 6454: 6436: 6435: 6434: 6421: 6400: 6398: 6397: 6392: 6390: 6389: 6380: 6379: 6378: 6365: 6357: 6352: 6334: 6333: 6287: 6285: 6284: 6279: 6271: 6263: 6249: 6247: 6246: 6241: 6239: 6232: 6231: 6216: 6211: 6191: 6188: 6183: 6161: 6159: 6158: 6157: 6132: 6126: 6125: 6106: 6101: 6079: 6077: 6076: 6075: 6050: 6044: 6043: 6033: 6028: 6006: 6004: 5990: 5974:geometric series 5968:Geometric series 5963: 5956: 5948: 5941: 5933: 5931: 5930: 5925: 5917: 5909: 5883: 5881: 5880: 5875: 5873: 5860: 5855: 5854: 5845: 5840: 5835: 5834: 5825: 5814: 5809: 5808: 5799: 5797: 5796: 5771: 5766: 5710: 5705: 5704: 5695: 5690: 5685: 5684: 5675: 5661: 5656: 5655: 5646: 5643: 5638: 5579: 5562: 5560: 5559: 5554: 5552: 5551: 5542: 5540: 5532: 5531: 5522: 5519: 5514: 5487: 5451:The exponential 5447: 5443: 5441: 5440: 5435: 5424: 5422: 5414: 5413: 5404: 5399: 5397: 5389: 5388: 5379: 5362: 5360: 5352: 5351: 5342: 5339: 5334: 5316: 5315: 5297: 5291: 5289: 5288: 5283: 5281: 5280: 5257: 5250: 5229: 5205: 5203: 5202: 5197: 5183: 5182: 5172: 5137: 5132: 5119: 5118: 5117: 5067: 5066: 5064: 5063: 5057: 5054: 5034: 5021: 5017: 4996: 4994: 4993: 4988: 4983: 4981: 4973: 4972: 4971: 4959: 4947: 4923: 4918: 4903: 4902: 4898: 4881: 4880: 4879: 4827: 4825: 4824: 4819: 4814: 4813: 4798: 4796: 4788: 4787: 4778: 4775: 4770: 4755: 4754: 4739: 4738: 4729: 4728: 4719: 4717: 4709: 4708: 4699: 4696: 4691: 4667: 4653: 4649: 4641: 4637: 4633: 4632: 4631: 4617: 4615: 4614: 4609: 4583: 4581: 4580: 4571: 4557: 4552: 4542: 4540: 4538: 4530: 4529: 4520: 4517: 4512: 4496: 4495: 4494: 4465: 4450: 4431: 4413: 4383: 4379: 4368:complex analysis 4361: 4346: 4321: 4310: 4299: 4292: 4281: 4268: 4266: 4265: 4260: 4258: 4257: 4242: 4239: 4219: 4216: 4212: 4211: 4210: 4209: 4200: 4152: 4141: 4136:smooth functions 4114: 4088: 4073: 4065: 4058: 4047: 4039: 4031: 4029: 4019: 4017: 4016: 4011: 4005: 4003: 3995: 3994: 3985: 3980: 3978: 3970: 3969: 3960: 3955: 3953: 3945: 3944: 3935: 3924: 3904: 3897: 3886: 3879: 3863: 3856: 3848: 3825:Taylor's theorem 3787:complex analysis 3751: 3747: 3743: 3735: 3731: 3703: 3690: 3686: 3672: 3666: 3664: 3663: 3658: 3656: 3655: 3643: 3641: 3633: 3623: 3622: 3606: 3595: 3585: 3581: 3575: 3573: 3572: 3567: 3562: 3561: 3540: 3539: 3529: 3524: 3485: 3482:in this region, 3481: 3473: 3469: 3454: 3447: 3396: 3394: 3393: 3388: 3383: 3374: 3368: 3367: 3342: 3340: 3339: 3334: 3329: 3328: 3322: 3321: 3312: 3307: 3305: 3304: 3280: 3274: 3272: 3271: 3266: 3264: 3263: 3262: 3247: 3238: 3235: 3234: 3227: 3218: 3199: 3197: 3195: 3194: 3189: 3176: 3174: 3173: 3168: 3147: 3145: 3144: 3139: 3119: 3117: 3116: 3111: 3091: 3089: 3088: 3083: 3058: 3056: 3054: 3053: 3048: 3026: 3024: 3023: 3018: 2998: 2996: 2995: 2990: 2970: 2969: 2967: 2966: 2961: 2857: 2850: 2842: 2836: 2830: 2827:with respect to 2826: 2818: 2816: 2815: 2810: 2808: 2795: 2790: 2789: 2780: 2775: 2770: 2769: 2760: 2755: 2750: 2749: 2740: 2735: 2730: 2729: 2720: 2700: 2690: 2688: 2680: 2679: 2670: 2665: 2663: 2655: 2654: 2645: 2640: 2638: 2630: 2629: 2620: 2615: 2613: 2605: 2604: 2595: 2590: 2588: 2580: 2579: 2570: 2565: 2563: 2555: 2554: 2545: 2536: 2534: 2526: 2525: 2516: 2513: 2508: 2480: 2469: 2467: 2466: 2461: 2450: 2445: 2444: 2439: 2435: 2419: 2417: 2415: 2414: 2402: 2382: 2374: 2351: 2347: 2338: 2336: 2335: 2330: 2319: 2318: 2297: 2288: 2282: 2281: 2260: 2251: 2245: 2244: 2223: 2214: 2184: 2177: 2168: 2166: 2165: 2160: 2149: 2148: 2139: 2130: 2124: 2123: 2114: 2105: 2099: 2098: 2089: 2080: 2055: 2051: 2041: 2039: 2038: 2033: 2022: 2021: 1997: 1996: 1942: 1935: 1934: 1932: 1931: 1926: 1923: 1915: 1908: 1902: 1900: 1899: 1894: 1883: 1882: 1870: 1869: 1839:geometric series 1836: 1835: 1833: 1832: 1826: 1823: 1801: 1799: 1798: 1793: 1788: 1787: 1778: 1776: 1768: 1758: 1757: 1741: 1738: 1733: 1709: 1708: 1699: 1697: 1689: 1679: 1670: 1665: 1664: 1655: 1653: 1645: 1635: 1626: 1618: 1616: 1608: 1598: 1589: 1565: 1551: 1547: 1535: 1531: 1527: 1523: 1515: 1511: 1500: 1492: 1485: 1483: 1482: 1477: 1472: 1471: 1450: 1448: 1440: 1430: 1429: 1413: 1410: 1405: 1381: 1380: 1359: 1357: 1349: 1339: 1330: 1325: 1324: 1303: 1301: 1293: 1283: 1274: 1254: 1252: 1244: 1234: 1225: 1198: 1181: 1155: 1139: 1118:Taylor's theorem 1115: 1107: 1103: 1095: 1076:Maclaurin series 1056:Taylor expansion 1036: 1029: 1022: 970: 935: 901: 900: 897: 864:Surface integral 807: 806: 803: 711: 710: 707: 667:Limit comparison 587: 586: 583: 469:Riemann integral 422: 421: 418: 378:L'Hôpital's rule 335:Taylor's theorem 256: 255: 252: 196: 194: 193: 188: 140: 131: 126: 96: 95: 91: 84: 78: 72: 66: 60: 54: 48: 42: 21: 18:Maclaurin series 20079: 20078: 20074: 20073: 20072: 20070: 20069: 20068: 20044: 20043: 20042: 20037: 20019: 19976: 19925:Kinds of series 19916: 19855: 19822:Explicit series 19813: 19787: 19749: 19735:Cauchy sequence 19723: 19710: 19664:Figurate number 19641: 19635: 19626:Powers of three 19570: 19561: 19519:"Taylor Series" 19497:"Taylor series" 19495: 19492: 19487: 19472: 19438: 19417: 19384: 19382: 19378: 19355: 19350: 19303: 19284: 19253: 19230: 19204:Hörmander, Lars 19188:10.2307/3028095 19151: 19120: 19106:Feller, William 19027: 19000: 18981: 18940: 18935: 18931: 18928:, p. 59–63 18915: 18911: 18899: 18892: 18880: 18876: 18860: 18856: 18844: 18840: 18828: 18824: 18816: 18812: 18808: 18786: 18779: 18767: 18763: 18755: 18751: 18747: 18731: 18727: 18719: 18715: 18711: 18707:Feigenbaum 1985 18688: 18684: 18680: 18660: 18656: 18648: 18644: 18632: 18628: 18616: 18612: 18604: 18600: 18592: 18588: 18576: 18572: 18568: 18523: 18516: 18513: 18468: 18444: 18438: 18428: 18417: 18386: 18380: 18354: 18346: 18330: 18326: 18314: 18269: 18265: 18263: 18260: 18259: 18249: 18247: 18246:is analytic in 18239: 18223: 18222: 18210: 18206: 18194: 18170: 18169: 18157: 18156: 18150: 18146: 18077: 18073: 18052: 18051: 18041: 17986: 17964: 17962: 17959: 17958: 17940: 17939: 17904: 17900: 17894: 17890: 17851: 17847: 17787: 17783: 17777: 17773: 17760: 17756: 17746: 17741: 17736: 17730: 17729: 17708: 17704: 17665: 17661: 17623: 17598: 17596: 17593: 17592: 17574: 17573: 17543: 17539: 17532: 17508: 17504: 17501: 17500: 17487: 17463: 17459: 17456: 17455: 17445: 17421: 17417: 17414: 17413: 17403: 17382: 17378: 17375: 17374: 17364: 17343: 17339: 17335: 17333: 17330: 17329: 17311: 17310: 17294: 17288: 17284: 17282: 17270: 17266: 17259: 17250: 17246: 17243: 17242: 17233: 17229: 17216: 17210: 17206: 17204: 17194: 17185: 17181: 17178: 17177: 17150: 17146: 17139: 17130: 17126: 17123: 17122: 17109: 17103: 17099: 17097: 17090: 17084: 17080: 17077: 17076: 17049: 17045: 17038: 17032: 17028: 17024: 17022: 17019: 17018: 16973: 16969: 16946: 16943: 16942: 16930: 16903: 16896: 16863: 16845: 16840: 16839: 16838: 16837: 16833: 16823: 16816: 16812: 16807: 16799: 16795: 16793: 16780: 16772: 16771: 16756: 16748: 16745: 16744: 16737:. Applying the 16720: 16710: 16706: 16688: 16658: 16650: 16634: 16622: 16618: 16617: 16613: 16606: 16605: 16601: 16596: 16588: 16575: 16570: 16559: 16543: 16542: 16538: 16533: 16525: 16511: 16494: 16486: 16483: 16482: 16477: 16471: 16443: 16442: 16418: 16414: 16408: 16404: 16365: 16361: 16301: 16297: 16291: 16287: 16269: 16268: 16258: 16253: 16229: 16225: 16186: 16182: 16144: 16141: 16140: 16126: 16122: 16118: 16087: 16084: 16083: 16067: 16066: 16051: 16047: 16038: 16034: 16022: 16018: 16009: 16005: 15993: 15989: 15980: 15976: 15964: 15960: 15951: 15947: 15938: 15934: 15930: 15920: 15916: 15901: 15897: 15885: 15881: 15880: 15878: 15872: 15861: 15851: 15840: 15830: 15819: 15805: 15800: 15789: 15788: 15779: 15775: 15766: 15762: 15750: 15746: 15737: 15733: 15721: 15717: 15708: 15704: 15700: 15690: 15686: 15671: 15667: 15655: 15651: 15650: 15648: 15642: 15631: 15621: 15610: 15596: 15591: 15579: 15575: 15566: 15562: 15550: 15546: 15542: 15532: 15528: 15513: 15509: 15499: 15497: 15491: 15480: 15464: 15460: 15445: 15441: 15426: 15425: 15416: 15412: 15397: 15393: 15375: 15371: 15370: 15365: 15347: 15343: 15342: 15337: 15329: 15317: 15313: 15298: 15294: 15293: 15289: 15288: 15286: 15282: 15269: 15265: 15253: 15249: 15248: 15239: 15235: 15234: 15230: 15224: 15220: 15211: 15207: 15193: 15189: 15188: 15184: 15178: 15174: 15165: 15161: 15157: 15155: 15149: 15136: 15132: 15131: 15118: 15105: 15101: 15100: 15089: 15080: 15076: 15061: 15057: 15047: 15045: 15042: 15041: 15036: 14982: 14964: 14963: 14954: 14950: 14940: 14929: 14927: 14921: 14910: 14897: 14896: 14890: 14886: 14876: 14865: 14863: 14857: 14846: 14827: 14826: 14820: 14816: 14789: 14784: 14771: 14766: 14765: 14761: 14755: 14744: 14709: 14703: 14699: 14697: 14691: 14680: 14663: 14657: 14653: 14651: 14645: 14634: 14615: 14614: 14604: 14592: 14588: 14586: 14580: 14569: 14552: 14546: 14542: 14540: 14534: 14523: 14500: 14488: 14484: 14482: 14476: 14465: 14448: 14442: 14438: 14436: 14430: 14419: 14406: 14402: 14390: 14386: 14379: 14373: 14369: 14350: 14348: 14345: 14344: 14310: 14304: 14300: 14298: 14285: 14279: 14275: 14273: 14260: 14254: 14250: 14248: 14223: 14217: 14213: 14211: 14205: 14194: 14181: 14177: 14175: 14172: 14171: 14163: 14159: 14148: 14145: 14115: 14111: 14099: 14090: 14086: 14074: 14065: 14061: 14033: 14027: 14023: 14021: 14019: 14016: 14015: 13990: 13987: 13986: 13985: 13968: 13964: 13962: 13959: 13958: 13933: 13930: 13929: 13912: 13908: 13906: 13903: 13902: 13865: 13856: 13852: 13840: 13831: 13827: 13815: 13806: 13802: 13781: 13772: 13768: 13756: 13747: 13743: 13722: 13713: 13709: 13697: 13688: 13684: 13663: 13659: 13638: 13634: 13632: 13629: 13628: 13607: 13603: 13601: 13598: 13597: 13563: 13560: 13559: 13543: 13542: 13530: 13526: 13511: 13505: 13501: 13499: 13485: 13481: 13479: 13470: 13466: 13465: 13461: 13452: 13448: 13432: 13428: 13426: 13417: 13413: 13412: 13408: 13399: 13395: 13379: 13375: 13373: 13364: 13360: 13359: 13355: 13343: 13339: 13330: 13326: 13317: 13316: 13295: 13289: 13285: 13283: 13270: 13264: 13260: 13258: 13251: 13247: 13230: 13226: 13220: 13216: 13207: 13203: 13197: 13193: 13184: 13180: 13174: 13170: 13158: 13154: 13145: 13141: 13140: 13136: 13129: 13123: 13119: 13115: 13113: 13110: 13109: 13085: 13082: 13081: 13056: 13052: 13046: 13042: 13033: 13029: 13023: 13019: 13010: 13006: 13000: 12996: 12984: 12980: 12971: 12967: 12951: 12945: 12941: 12939: 12937: 12934: 12933: 12899: 12893: 12889: 12887: 12874: 12868: 12864: 12862: 12842: 12839: 12838: 12804: 12798: 12794: 12792: 12779: 12773: 12769: 12767: 12754: 12748: 12744: 12742: 12721: 12717: 12715: 12712: 12711: 12679: 12673: 12669: 12667: 12650: 12647: 12646: 12641: 12615: 12614: 12600: 12596: 12592: 12591: 12574: 12570: 12568: 12554: 12550: 12548: 12534: 12530: 12528: 12516: 12515: 12503: 12499: 12480: 12476: 12475: 12463: 12459: 12429: 12420: 12416: 12386: 12353: 12352: 12346: 12345: 12312: 12311: 12298: 12282: 12280: 12277: 12276: 12246: 12242: 12238: 12237: 12220: 12216: 12214: 12200: 12196: 12194: 12180: 12176: 12174: 12151: 12148: 12147: 12121: 12117: 12113: 12112: 12095: 12091: 12089: 12075: 12071: 12069: 12037: 12034: 12033: 11993: 11990: 11989: 11949: 11946: 11945: 11923: 11922: 11889: 11888: 11883: 11866: 11863: 11862: 11838: 11837: 11825: 11809: 11805: 11799: 11798: 11750: 11747: 11746: 11741: 11724: 11691: 11687: 11680: 11675: 11669: 11658: 11645: 11641: 11623: 11612: 11595: 11591: 11585: 11584: 11583: 11572: 11565: 11561: 11549: 11545: 11536: 11532: 11520: 11516: 11515: 11513: 11507: 11506: 11496: 11489: 11485: 11473: 11469: 11459: 11455: 11451: 11439: 11435: 11433: 11430: 11429: 11405: 11401: 11394: 11389: 11383: 11372: 11359: 11355: 11337: 11326: 11309: 11302: 11296: 11295: 11294: 11283: 11276: 11272: 11260: 11256: 11247: 11243: 11231: 11227: 11226: 11224: 11218: 11217: 11207: 11200: 11196: 11184: 11180: 11170: 11163: 11159: 11147: 11143: 11141: 11138: 11137: 11115: 11114: 11106: 11102: 11101: 11097: 11091: 11087: 11072: 11061: 11041: 11026: 11022: 11019: 11018: 11010: 11006: 11005: 11001: 10995: 10984: 10964: 10949: 10945: 10941: 10939: 10936: 10935: 10913: 10912: 10903: 10899: 10890: 10886: 10871: 10867: 10848: 10841: 10837: 10818: 10816: 10810: 10799: 10788: 10766: 10763: 10762: 10753: 10749: 10740: 10736: 10721: 10717: 10716: 10709: 10705: 10686: 10684: 10678: 10667: 10656: 10634: 10630: 10628: 10625: 10624: 10615: 10590: 10589: 10574: 10570: 10561: 10557: 10541: 10534: 10530: 10520: 10518: 10512: 10501: 10490: 10475: 10470: 10469: 10466: 10465: 10450: 10446: 10437: 10433: 10417: 10410: 10406: 10396: 10394: 10388: 10377: 10366: 10351: 10346: 10345: 10341: 10339: 10336: 10335: 10311: 10310: 10295: 10291: 10282: 10278: 10262: 10257: 10251: 10240: 10229: 10214: 10210: 10207: 10206: 10191: 10187: 10178: 10174: 10158: 10153: 10147: 10136: 10125: 10110: 10106: 10102: 10100: 10097: 10096: 10074: 10073: 10067: 10063: 10055: 10051: 10046: 10040: 10029: 10018: 10003: 9998: 9997: 9994: 9993: 9987: 9983: 9975: 9971: 9966: 9960: 9949: 9938: 9923: 9918: 9917: 9913: 9911: 9908: 9907: 9898: 9882: 9878: 9873: 9857: 9856: 9827: 9819: 9814: 9811: 9794: 9790: 9788: 9774: 9770: 9768: 9754: 9738: 9723: 9719: 9717: 9711: 9700: 9689: 9677: 9676: 9665: 9657: 9652: 9649: 9631: 9627: 9623: 9621: 9607: 9603: 9601: 9587: 9572: 9568: 9541: 9537: 9522: 9518: 9517: 9495: 9491: 9481: 9479: 9473: 9462: 9451: 9439: 9438: 9428: 9420: 9412: 9407: 9404: 9386: 9382: 9378: 9376: 9361: 9357: 9353: 9351: 9337: 9333: 9331: 9317: 9302: 9298: 9279: 9261: 9257: 9256: 9252: 9246: 9242: 9233: 9229: 9228: 9226: 9220: 9209: 9198: 9186: 9185: 9177: 9174: 9158: 9152: 9148: 9146: 9133: 9127: 9123: 9121: 9107: 9088: 9079: 9075: 9073: 9067: 9056: 9045: 9033: 9032: 9024: 9021: 9005: 8999: 8995: 8993: 8980: 8974: 8970: 8968: 8954: 8929: 8914: 8910: 8908: 8902: 8891: 8880: 8867: 8865: 8862: 8861: 8852: 8836: 8833: 8825: 8814: 8810: 8805: 8785: 8784: 8755: 8747: 8742: 8739: 8722: 8718: 8716: 8702: 8698: 8696: 8682: 8667: 8663: 8647: 8640: 8636: 8626: 8624: 8618: 8607: 8596: 8584: 8583: 8572: 8564: 8559: 8556: 8538: 8534: 8530: 8528: 8514: 8510: 8508: 8489: 8481: 8466: 8462: 8435: 8431: 8416: 8412: 8411: 8394: 8392: 8386: 8375: 8358: 8349: 8348: 8326: 8319: 8307: 8306: 8295: 8287: 8282: 8279: 8261: 8257: 8253: 8251: 8237: 8233: 8231: 8217: 8202: 8198: 8171: 8167: 8152: 8148: 8147: 8130: 8128: 8122: 8111: 8100: 8088: 8087: 8077: 8069: 8061: 8056: 8053: 8035: 8031: 8027: 8025: 8011: 8007: 8005: 7991: 7982: 7978: 7959: 7949: 7945: 7939: 7935: 7925: 7923: 7917: 7906: 7895: 7883: 7882: 7872: 7864: 7856: 7851: 7848: 7830: 7826: 7822: 7820: 7806: 7802: 7800: 7786: 7771: 7767: 7748: 7736: 7732: 7725: 7721: 7715: 7711: 7693: 7689: 7688: 7686: 7680: 7669: 7658: 7646: 7645: 7637: 7634: 7618: 7612: 7608: 7606: 7593: 7587: 7583: 7581: 7567: 7558: 7554: 7535: 7528: 7524: 7514: 7512: 7506: 7495: 7484: 7472: 7471: 7463: 7460: 7444: 7438: 7434: 7432: 7419: 7413: 7409: 7407: 7393: 7378: 7374: 7349: 7342: 7338: 7328: 7326: 7320: 7309: 7298: 7285: 7283: 7280: 7279: 7270: 7241: 7240: 7231: 7227: 7218: 7214: 7199: 7195: 7194: 7172: 7168: 7158: 7156: 7150: 7139: 7130: 7118: 7114: 7104: 7095: 7091: 7081: 7072: 7068: 7058: 7049: 7045: 7035: 7019: 7006: 6994: 6990: 6986: 6971: 6970: 6961: 6957: 6930: 6926: 6911: 6907: 6906: 6878: 6874: 6864: 6862: 6856: 6845: 6836: 6824: 6820: 6810: 6801: 6797: 6787: 6778: 6774: 6764: 6755: 6751: 6741: 6725: 6712: 6701: 6697: 6681: 6679: 6676: 6675: 6657: 6654: 6651: 6650: 6648: 6643: 6635: 6632: 6629: 6628: 6626: 6621: 6614: 6607: 6585: 6577: 6575: 6572: 6571: 6560: 6534: 6490: 6488: 6463: 6461: 6455: 6444: 6430: 6417: 6416: 6415: 6413: 6410: 6409: 6385: 6381: 6374: 6361: 6360: 6359: 6353: 6342: 6329: 6325: 6311: 6308: 6307: 6302:binomial series 6298: 6296:Binomial series 6290:binomial series 6267: 6259: 6257: 6254: 6253: 6237: 6236: 6221: 6217: 6192: 6190: 6184: 6173: 6162: 6153: 6149: 6136: 6131: 6128: 6127: 6115: 6111: 6102: 6091: 6080: 6071: 6067: 6054: 6049: 6046: 6045: 6039: 6035: 6029: 6018: 6007: 5994: 5989: 5985: 5983: 5980: 5979: 5970: 5958: 5950: 5943: 5935: 5913: 5905: 5903: 5900: 5899: 5888:Mercator series 5871: 5870: 5850: 5846: 5844: 5830: 5826: 5824: 5804: 5800: 5798: 5786: 5782: 5767: 5756: 5745: 5721: 5720: 5700: 5696: 5694: 5680: 5676: 5674: 5651: 5647: 5645: 5639: 5628: 5614: 5589: 5587: 5584: 5583: 5575: 5568: 5547: 5543: 5533: 5527: 5523: 5521: 5515: 5504: 5483: 5466: 5463: 5462: 5445: 5415: 5409: 5405: 5403: 5390: 5384: 5380: 5378: 5353: 5347: 5343: 5341: 5335: 5324: 5311: 5307: 5305: 5302: 5301: 5293: 5276: 5272: 5270: 5267: 5266: 5252: 5246: 5236: 5227: 5224: 5218: 5168: 5164: 5160: 5133: 5125: 5113: 5109: 5102: 5075: 5072: 5071: 5058: 5055: 5044: 5043: 5041: 5036: 5030: 5027:random variable 5019: 5004: 4974: 4967: 4963: 4955: 4948: 4946: 4919: 4908: 4894: 4887: 4883: 4875: 4871: 4864: 4837: 4834: 4833: 4803: 4799: 4789: 4783: 4779: 4777: 4771: 4760: 4747: 4743: 4734: 4730: 4724: 4720: 4710: 4704: 4700: 4698: 4692: 4681: 4675: 4672: 4671: 4666: 4658: 4651: 4650:is analytic at 4647: 4639: 4635: 4630: 4625: 4624: 4623: 4621: 4576: 4572: 4553: 4548: 4543: 4541: 4531: 4525: 4521: 4519: 4513: 4502: 4490: 4486: 4479: 4473: 4470: 4469: 4460: 4459:, that for any 4448: 4438: 4419: 4418:. For example, 4411: 4381: 4375: 4352: 4337: 4312: 4301: 4294: 4283: 4276: 4253: 4252: 4238: 4236: 4230: 4229: 4215: 4213: 4205: 4201: 4196: 4189: 4185: 4178: 4177: 4160: 4157: 4156: 4143: 4139: 4108: 4100: 4086: 4067: 4060: 4052: 4041: 4033: 4025: 4023: 3996: 3990: 3986: 3984: 3971: 3965: 3961: 3959: 3946: 3940: 3936: 3934: 3920: 3912: 3909: 3908: 3899: 3892: 3881: 3873: 3858: 3850: 3842: 3827: 3821: 3805:Euler's formula 3749: 3745: 3741: 3733: 3729: 3699: 3688: 3677: 3670: 3651: 3647: 3634: 3612: 3608: 3607: 3605: 3603: 3600: 3599: 3587: 3583: 3579: 3557: 3553: 3535: 3531: 3525: 3514: 3493: 3490: 3489: 3483: 3479: 3471: 3460: 3449: 3441: 3434: 3428: 3420:Colin Maclaurin 3372: 3363: 3359: 3348: 3345: 3344: 3324: 3323: 3317: 3313: 3306: 3300: 3299: 3291: 3288: 3287: 3278: 3258: 3257: 3236: 3230: 3229: 3228: 3216: 3204: 3201: 3200: 3183: 3180: 3179: 3178: 3152: 3149: 3148: 3124: 3121: 3120: 3096: 3093: 3092: 3068: 3065: 3064: 3033: 3030: 3029: 3028: 3003: 3000: 2999: 2975: 2972: 2971: 2946: 2943: 2942: 2940: 2864: 2852: 2844: 2838: 2832: 2828: 2822: 2806: 2805: 2785: 2781: 2779: 2765: 2761: 2759: 2745: 2741: 2739: 2725: 2721: 2719: 2698: 2697: 2681: 2675: 2671: 2669: 2656: 2650: 2646: 2644: 2631: 2625: 2621: 2619: 2606: 2600: 2596: 2594: 2581: 2575: 2571: 2569: 2556: 2550: 2546: 2544: 2537: 2527: 2521: 2517: 2515: 2509: 2498: 2490: 2488: 2485: 2484: 2476: 2440: 2425: 2421: 2420: 2418: 2410: 2406: 2401: 2373: 2359: 2356: 2355: 2349: 2342: 2314: 2310: 2286: 2277: 2273: 2249: 2240: 2236: 2212: 2192: 2189: 2188: 2179: 2172: 2144: 2140: 2128: 2119: 2115: 2103: 2094: 2090: 2078: 2067: 2064: 2063: 2053: 2045: 2017: 2013: 1992: 1988: 1950: 1947: 1946: 1937: 1927: 1924: 1921: 1920: 1918: 1917: 1910: 1906: 1878: 1874: 1865: 1861: 1847: 1844: 1843: 1827: 1824: 1821: 1820: 1818: 1817: 1807: 1783: 1779: 1769: 1747: 1743: 1742: 1740: 1734: 1723: 1704: 1700: 1690: 1672: 1671: 1669: 1660: 1656: 1646: 1628: 1627: 1625: 1609: 1591: 1590: 1588: 1571: 1568: 1567: 1560: 1549: 1537: 1533: 1529: 1525: 1521: 1513: 1502: 1501:. The function 1498: 1487: 1467: 1463: 1441: 1419: 1415: 1414: 1412: 1406: 1395: 1376: 1372: 1350: 1332: 1331: 1329: 1320: 1316: 1294: 1276: 1275: 1273: 1245: 1227: 1226: 1224: 1207: 1204: 1203: 1194: 1172: 1162: 1153: 1137: 1113: 1105: 1101: 1090: 1080:Colin Maclaurin 1040: 1011: 1010: 996:Integration Bee 971: 968: 961: 960: 936: 933: 926: 925: 898: 895: 888: 887: 869:Volume integral 804: 799: 792: 791: 708: 703: 696: 695: 665: 584: 579: 572: 571: 563:Risch algorithm 533:Euler's formula 419: 414: 407: 406: 388:General Leibniz 271:generalizations 253: 248: 241: 227:Rolle's theorem 222: 197: 133: 127: 122: 116: 113: 112: 86: 80: 74: 68: 62: 56: 50: 44: 37: 28: 23: 22: 15: 12: 11: 5: 20077: 20067: 20066: 20061: 20056: 20039: 20038: 20036: 20035: 20024: 20021: 20020: 20018: 20017: 20012: 20007: 20002: 19997: 19992: 19986: 19984: 19978: 19977: 19975: 19974: 19969: 19967:Fourier series 19964: 19959: 19954: 19952:Puiseux series 19949: 19947:Laurent series 19944: 19939: 19934: 19928: 19926: 19922: 19921: 19918: 19917: 19915: 19914: 19909: 19904: 19899: 19894: 19889: 19884: 19879: 19874: 19869: 19863: 19861: 19857: 19856: 19854: 19853: 19848: 19843: 19838: 19832: 19830: 19823: 19819: 19818: 19815: 19814: 19812: 19811: 19806: 19801: 19795: 19793: 19789: 19788: 19786: 19785: 19780: 19775: 19770: 19764: 19762: 19755: 19751: 19750: 19748: 19747: 19742: 19737: 19731: 19729: 19725: 19724: 19717: 19715: 19712: 19711: 19709: 19708: 19707: 19706: 19696: 19691: 19686: 19681: 19676: 19671: 19666: 19661: 19656: 19650: 19648: 19637: 19636: 19634: 19633: 19628: 19623: 19618: 19613: 19608: 19603: 19598: 19593: 19587: 19585: 19578: 19572: 19571: 19560: 19559: 19552: 19545: 19537: 19531: 19530: 19511: 19491: 19490:External links 19488: 19486: 19485: 19476: 19470: 19457: 19442: 19436: 19421: 19415: 19399: 19390: 19366:(5): 291–306. 19344: 19307: 19301: 19288: 19282: 19268:. Birkhauser. 19257: 19251: 19234: 19228: 19200: 19171: 19155: 19149: 19132: 19118: 19102: 19084:(1–2): 1–140. 19073: 19055:(3): 236–246. 19044: 19031: 19025: 19004: 18998: 18985: 18979: 18964: 18941: 18939: 18936: 18934: 18933: 18930: 18929: 18923: 18920:Hörmander 2002 18916: 18909: 18890: 18874: 18854: 18838: 18822: 18810: 18807: 18806: 18797: 18787: 18777: 18761: 18759:, p. 231. 18749: 18746: 18745: 18739: 18732: 18725: 18713: 18710: 18709: 18704: 18689: 18682: 18679: 18678: 18673: 18668: 18661: 18654: 18642: 18626: 18610: 18598: 18586: 18569: 18567: 18564: 18563: 18562: 18557: 18552: 18549:Puiseux series 18546: 18540: 18535: 18529: 18528: 18512: 18509: 18508: 18507: 18503: 18500:quadratic mean 18483: 18461: 18450: 18390:Fourier series 18384:Fourier series 18382:Main article: 18379: 18376: 18364: 18361: 18357: 18353: 18349: 18344: 18341: 18338: 18333: 18329: 18322: 18319: 18313: 18310: 18307: 18304: 18301: 18298: 18295: 18292: 18289: 18286: 18283: 18280: 18277: 18272: 18268: 18221: 18218: 18213: 18209: 18202: 18199: 18193: 18190: 18187: 18184: 18181: 18178: 18175: 18173: 18171: 18168: 18165: 18160: 18153: 18149: 18145: 18142: 18139: 18136: 18133: 18130: 18127: 18124: 18121: 18118: 18115: 18112: 18109: 18106: 18103: 18100: 18097: 18094: 18091: 18088: 18085: 18080: 18076: 18072: 18069: 18066: 18063: 18060: 18055: 18048: 18045: 18040: 18037: 18034: 18031: 18028: 18025: 18022: 18019: 18016: 18013: 18010: 18007: 18004: 18001: 17998: 17995: 17992: 17989: 17987: 17985: 17982: 17979: 17976: 17973: 17970: 17967: 17966: 17938: 17935: 17931: 17927: 17924: 17921: 17918: 17915: 17910: 17907: 17903: 17897: 17893: 17889: 17886: 17883: 17880: 17877: 17874: 17871: 17868: 17865: 17862: 17857: 17854: 17850: 17846: 17843: 17840: 17837: 17834: 17831: 17828: 17825: 17822: 17819: 17816: 17813: 17810: 17807: 17804: 17801: 17798: 17793: 17790: 17786: 17780: 17776: 17772: 17769: 17766: 17763: 17759: 17752: 17749: 17745: 17740: 17735: 17733: 17731: 17728: 17725: 17722: 17719: 17716: 17711: 17707: 17703: 17700: 17697: 17694: 17691: 17688: 17685: 17682: 17679: 17676: 17673: 17668: 17664: 17660: 17657: 17654: 17651: 17648: 17645: 17642: 17639: 17636: 17633: 17630: 17627: 17624: 17622: 17619: 17616: 17613: 17610: 17607: 17604: 17601: 17600: 17572: 17569: 17566: 17563: 17560: 17557: 17554: 17549: 17546: 17542: 17538: 17535: 17533: 17531: 17528: 17525: 17522: 17519: 17514: 17511: 17507: 17503: 17502: 17499: 17496: 17493: 17490: 17488: 17486: 17483: 17480: 17477: 17474: 17469: 17466: 17462: 17458: 17457: 17454: 17451: 17448: 17446: 17444: 17441: 17438: 17435: 17432: 17427: 17424: 17420: 17416: 17415: 17412: 17409: 17406: 17404: 17402: 17399: 17396: 17393: 17390: 17385: 17381: 17377: 17376: 17373: 17370: 17367: 17365: 17363: 17360: 17357: 17354: 17351: 17346: 17342: 17338: 17337: 17309: 17303: 17300: 17297: 17291: 17287: 17281: 17276: 17273: 17269: 17265: 17262: 17260: 17256: 17253: 17249: 17245: 17244: 17236: 17232: 17228: 17225: 17222: 17219: 17213: 17209: 17203: 17200: 17197: 17195: 17191: 17188: 17184: 17180: 17179: 17176: 17173: 17170: 17167: 17164: 17161: 17158: 17153: 17149: 17145: 17142: 17140: 17136: 17133: 17129: 17125: 17124: 17118: 17115: 17112: 17106: 17102: 17096: 17093: 17091: 17087: 17083: 17079: 17078: 17075: 17072: 17069: 17066: 17063: 17060: 17057: 17052: 17048: 17044: 17041: 17039: 17035: 17031: 17027: 17026: 17002: 16999: 16996: 16993: 16990: 16987: 16984: 16981: 16976: 16972: 16968: 16965: 16962: 16959: 16956: 16953: 16950: 16895: 16892: 16873: 16870: 16866: 16862: 16858: 16854: 16848: 16843: 16836: 16829: 16826: 16819: 16815: 16810: 16806: 16802: 16798: 16790: 16787: 16783: 16779: 16775: 16770: 16766: 16763: 16759: 16755: 16752: 16735:Hessian matrix 16674: 16671: 16668: 16665: 16661: 16657: 16653: 16649: 16645: 16641: 16637: 16633: 16630: 16625: 16621: 16616: 16609: 16604: 16599: 16595: 16591: 16587: 16581: 16578: 16574: 16569: 16566: 16562: 16558: 16555: 16552: 16546: 16541: 16536: 16532: 16528: 16524: 16521: 16518: 16514: 16510: 16507: 16504: 16501: 16497: 16493: 16490: 16470: 16467: 16446: 16441: 16438: 16435: 16432: 16429: 16424: 16421: 16417: 16411: 16407: 16403: 16400: 16397: 16394: 16391: 16388: 16385: 16382: 16379: 16376: 16371: 16368: 16364: 16360: 16357: 16354: 16351: 16348: 16345: 16342: 16339: 16336: 16333: 16330: 16327: 16324: 16321: 16318: 16315: 16312: 16307: 16304: 16300: 16294: 16290: 16286: 16283: 16280: 16277: 16272: 16264: 16261: 16257: 16252: 16249: 16246: 16243: 16240: 16237: 16232: 16228: 16224: 16221: 16218: 16215: 16212: 16209: 16206: 16203: 16200: 16197: 16194: 16189: 16185: 16181: 16178: 16175: 16172: 16169: 16166: 16163: 16160: 16157: 16154: 16151: 16148: 16106: 16103: 16100: 16097: 16094: 16091: 16065: 16062: 16059: 16054: 16050: 16046: 16041: 16037: 16033: 16030: 16025: 16021: 16017: 16012: 16008: 16004: 16001: 15996: 15992: 15988: 15983: 15979: 15975: 15967: 15963: 15959: 15954: 15950: 15946: 15941: 15937: 15933: 15928: 15923: 15919: 15915: 15912: 15909: 15904: 15900: 15896: 15893: 15888: 15884: 15875: 15870: 15867: 15864: 15860: 15854: 15849: 15846: 15843: 15839: 15833: 15828: 15825: 15822: 15818: 15811: 15808: 15804: 15799: 15794: 15792: 15790: 15787: 15782: 15778: 15774: 15769: 15765: 15761: 15758: 15753: 15749: 15745: 15740: 15736: 15732: 15724: 15720: 15716: 15711: 15707: 15703: 15698: 15693: 15689: 15685: 15682: 15679: 15674: 15670: 15666: 15663: 15658: 15654: 15645: 15640: 15637: 15634: 15630: 15624: 15619: 15616: 15613: 15609: 15602: 15599: 15595: 15590: 15587: 15582: 15578: 15574: 15569: 15565: 15561: 15553: 15549: 15545: 15540: 15535: 15531: 15527: 15524: 15521: 15516: 15512: 15508: 15505: 15502: 15494: 15489: 15486: 15483: 15479: 15475: 15472: 15467: 15463: 15459: 15456: 15453: 15448: 15444: 15440: 15437: 15434: 15431: 15429: 15427: 15424: 15419: 15415: 15411: 15408: 15405: 15400: 15396: 15392: 15388: 15378: 15374: 15368: 15364: 15360: 15357: 15350: 15346: 15340: 15336: 15332: 15327: 15320: 15316: 15312: 15309: 15306: 15301: 15297: 15292: 15285: 15277: 15272: 15268: 15264: 15261: 15256: 15252: 15242: 15238: 15233: 15227: 15223: 15219: 15214: 15210: 15206: 15203: 15196: 15192: 15187: 15181: 15177: 15173: 15168: 15164: 15160: 15152: 15147: 15144: 15139: 15135: 15130: 15126: 15121: 15116: 15113: 15108: 15104: 15099: 15095: 15092: 15090: 15088: 15083: 15079: 15075: 15072: 15069: 15064: 15060: 15056: 15053: 15050: 15049: 15035: 15032: 14981: 14978: 14962: 14957: 14953: 14946: 14943: 14938: 14935: 14932: 14924: 14919: 14916: 14913: 14909: 14905: 14902: 14900: 14898: 14893: 14889: 14882: 14879: 14874: 14871: 14868: 14860: 14855: 14852: 14849: 14845: 14841: 14838: 14835: 14832: 14830: 14828: 14823: 14819: 14814: 14807: 14804: 14801: 14798: 14795: 14792: 14788: 14783: 14777: 14774: 14770: 14764: 14758: 14753: 14750: 14747: 14743: 14739: 14736: 14733: 14727: 14724: 14721: 14718: 14715: 14712: 14706: 14702: 14694: 14689: 14686: 14683: 14679: 14675: 14669: 14666: 14660: 14656: 14648: 14643: 14640: 14637: 14633: 14629: 14626: 14623: 14620: 14618: 14616: 14610: 14607: 14601: 14598: 14595: 14591: 14583: 14578: 14575: 14572: 14568: 14564: 14558: 14555: 14549: 14545: 14537: 14532: 14529: 14526: 14522: 14518: 14515: 14512: 14506: 14503: 14497: 14494: 14491: 14487: 14479: 14474: 14471: 14468: 14464: 14460: 14454: 14451: 14445: 14441: 14433: 14428: 14425: 14422: 14418: 14414: 14409: 14405: 14401: 14398: 14393: 14389: 14385: 14382: 14380: 14376: 14372: 14368: 14365: 14362: 14359: 14356: 14353: 14352: 14328: 14325: 14322: 14316: 14313: 14307: 14303: 14297: 14291: 14288: 14282: 14278: 14272: 14266: 14263: 14257: 14253: 14247: 14244: 14241: 14238: 14235: 14229: 14226: 14220: 14216: 14208: 14203: 14200: 14197: 14193: 14189: 14184: 14180: 14144: 14141: 14129: 14126: 14123: 14118: 14114: 14107: 14104: 14098: 14093: 14089: 14082: 14079: 14073: 14068: 14064: 14060: 14057: 14054: 14051: 14048: 14042: 14039: 14036: 14030: 14026: 14000: 13997: 13994: 13971: 13967: 13946: 13943: 13940: 13937: 13915: 13911: 13888: 13885: 13879: 13873: 13870: 13864: 13859: 13855: 13848: 13845: 13839: 13834: 13830: 13823: 13820: 13814: 13809: 13805: 13795: 13789: 13786: 13780: 13775: 13771: 13764: 13761: 13755: 13750: 13746: 13736: 13730: 13727: 13721: 13716: 13712: 13705: 13702: 13696: 13691: 13687: 13677: 13674: 13671: 13666: 13662: 13652: 13649: 13646: 13641: 13637: 13615: 13610: 13606: 13585: 13582: 13579: 13576: 13573: 13570: 13567: 13541: 13538: 13533: 13529: 13524: 13517: 13514: 13508: 13504: 13498: 13493: 13488: 13484: 13478: 13473: 13469: 13464: 13460: 13455: 13451: 13446: 13440: 13435: 13431: 13425: 13420: 13416: 13411: 13407: 13402: 13398: 13393: 13387: 13382: 13378: 13372: 13367: 13363: 13358: 13354: 13351: 13346: 13342: 13338: 13333: 13329: 13325: 13322: 13320: 13318: 13314: 13310: 13307: 13301: 13298: 13292: 13288: 13282: 13276: 13273: 13267: 13263: 13257: 13254: 13250: 13245: 13241: 13238: 13233: 13229: 13223: 13219: 13215: 13210: 13206: 13200: 13196: 13192: 13187: 13183: 13177: 13173: 13169: 13166: 13161: 13157: 13153: 13148: 13144: 13139: 13135: 13132: 13130: 13126: 13122: 13118: 13117: 13095: 13092: 13089: 13067: 13064: 13059: 13055: 13049: 13045: 13041: 13036: 13032: 13026: 13022: 13018: 13013: 13009: 13003: 12999: 12995: 12992: 12987: 12983: 12979: 12974: 12970: 12966: 12960: 12957: 12954: 12948: 12944: 12917: 12914: 12911: 12905: 12902: 12896: 12892: 12886: 12880: 12877: 12871: 12867: 12861: 12858: 12855: 12852: 12849: 12846: 12822: 12819: 12816: 12810: 12807: 12801: 12797: 12791: 12785: 12782: 12776: 12772: 12766: 12760: 12757: 12751: 12747: 12741: 12738: 12735: 12732: 12729: 12724: 12720: 12694: 12688: 12685: 12682: 12676: 12672: 12666: 12663: 12660: 12657: 12654: 12640: 12639:Second example 12637: 12613: 12608: 12603: 12599: 12595: 12590: 12587: 12582: 12577: 12573: 12567: 12562: 12557: 12553: 12547: 12542: 12537: 12533: 12527: 12524: 12521: 12519: 12517: 12512: 12506: 12502: 12498: 12495: 12492: 12489: 12486: 12483: 12479: 12474: 12471: 12466: 12462: 12458: 12455: 12452: 12449: 12446: 12443: 12437: 12434: 12428: 12423: 12419: 12415: 12412: 12409: 12406: 12403: 12400: 12394: 12391: 12385: 12382: 12379: 12376: 12373: 12370: 12367: 12364: 12361: 12358: 12356: 12354: 12349: 12344: 12341: 12338: 12335: 12332: 12329: 12326: 12323: 12320: 12315: 12310: 12307: 12304: 12301: 12299: 12297: 12294: 12291: 12288: 12285: 12284: 12259: 12254: 12249: 12245: 12241: 12236: 12233: 12228: 12223: 12219: 12213: 12208: 12203: 12199: 12193: 12188: 12183: 12179: 12173: 12170: 12167: 12164: 12161: 12158: 12155: 12129: 12124: 12120: 12116: 12111: 12108: 12103: 12098: 12094: 12088: 12083: 12078: 12074: 12068: 12065: 12062: 12059: 12056: 12053: 12050: 12047: 12044: 12041: 12028:big O notation 12015: 12012: 12009: 12006: 12003: 12000: 11997: 11977: 11974: 11971: 11968: 11965: 11962: 11959: 11956: 11953: 11931: 11926: 11921: 11918: 11915: 11912: 11909: 11906: 11903: 11900: 11897: 11892: 11886: 11882: 11879: 11876: 11873: 11870: 11846: 11841: 11833: 11830: 11824: 11817: 11814: 11808: 11802: 11797: 11794: 11790: 11787: 11784: 11781: 11778: 11775: 11772: 11769: 11766: 11763: 11760: 11757: 11754: 11740: 11737: 11723: 11720: 11703: 11700: 11697: 11694: 11690: 11686: 11683: 11679: 11672: 11667: 11664: 11661: 11657: 11653: 11648: 11644: 11640: 11637: 11634: 11631: 11626: 11621: 11618: 11615: 11611: 11607: 11602: 11598: 11594: 11588: 11579: 11575: 11568: 11564: 11560: 11557: 11552: 11548: 11544: 11539: 11535: 11531: 11528: 11523: 11519: 11510: 11503: 11499: 11495: 11492: 11488: 11484: 11481: 11476: 11472: 11466: 11462: 11458: 11454: 11450: 11447: 11442: 11438: 11408: 11404: 11400: 11397: 11393: 11386: 11381: 11378: 11375: 11371: 11367: 11362: 11358: 11354: 11351: 11348: 11345: 11340: 11335: 11332: 11329: 11325: 11321: 11316: 11312: 11308: 11305: 11299: 11290: 11286: 11279: 11275: 11271: 11268: 11263: 11259: 11255: 11250: 11246: 11242: 11239: 11234: 11230: 11221: 11214: 11210: 11206: 11203: 11199: 11195: 11192: 11187: 11183: 11177: 11173: 11169: 11166: 11162: 11158: 11155: 11150: 11146: 11109: 11105: 11100: 11094: 11090: 11086: 11083: 11080: 11075: 11070: 11067: 11064: 11060: 11056: 11053: 11050: 11047: 11044: 11042: 11040: 11037: 11034: 11029: 11025: 11021: 11020: 11013: 11009: 11004: 10998: 10993: 10990: 10987: 10983: 10979: 10976: 10973: 10970: 10967: 10965: 10963: 10960: 10957: 10952: 10948: 10944: 10943: 10909: 10906: 10902: 10893: 10889: 10885: 10882: 10879: 10874: 10870: 10866: 10863: 10860: 10857: 10854: 10851: 10844: 10840: 10836: 10833: 10830: 10827: 10824: 10821: 10813: 10808: 10805: 10802: 10798: 10794: 10791: 10789: 10787: 10784: 10781: 10778: 10773: 10770: 10765: 10764: 10759: 10756: 10752: 10743: 10739: 10735: 10732: 10729: 10724: 10720: 10712: 10708: 10704: 10701: 10698: 10695: 10692: 10689: 10681: 10676: 10673: 10670: 10666: 10662: 10659: 10657: 10655: 10652: 10649: 10646: 10641: 10638: 10633: 10632: 10614: 10611: 10586: 10583: 10580: 10577: 10573: 10564: 10560: 10556: 10553: 10550: 10547: 10544: 10537: 10533: 10529: 10526: 10523: 10515: 10510: 10507: 10504: 10500: 10496: 10493: 10491: 10489: 10486: 10483: 10478: 10468: 10467: 10462: 10459: 10456: 10453: 10449: 10440: 10436: 10432: 10429: 10426: 10423: 10420: 10413: 10409: 10405: 10402: 10399: 10391: 10386: 10383: 10380: 10376: 10372: 10369: 10367: 10365: 10362: 10359: 10354: 10344: 10343: 10307: 10304: 10301: 10298: 10294: 10285: 10281: 10277: 10274: 10271: 10268: 10265: 10261: 10254: 10249: 10246: 10243: 10239: 10235: 10232: 10230: 10228: 10225: 10222: 10217: 10213: 10209: 10208: 10203: 10200: 10197: 10194: 10190: 10181: 10177: 10173: 10170: 10167: 10164: 10161: 10157: 10150: 10145: 10142: 10139: 10135: 10131: 10128: 10126: 10124: 10121: 10118: 10113: 10109: 10105: 10104: 10070: 10066: 10058: 10054: 10050: 10043: 10038: 10035: 10032: 10028: 10024: 10021: 10019: 10017: 10014: 10011: 10006: 9996: 9995: 9990: 9986: 9978: 9974: 9970: 9963: 9958: 9955: 9952: 9948: 9944: 9941: 9939: 9937: 9934: 9931: 9926: 9916: 9915: 9902:polylogarithms 9897: 9894: 9876: 9855: 9852: 9849: 9846: 9840: 9837: 9834: 9830: 9826: 9822: 9813: 9810: 9807: 9802: 9797: 9793: 9787: 9782: 9777: 9773: 9767: 9764: 9761: 9758: 9756: 9750: 9747: 9744: 9741: 9735: 9732: 9729: 9726: 9722: 9714: 9709: 9706: 9703: 9699: 9695: 9692: 9690: 9688: 9685: 9682: 9679: 9678: 9675: 9672: 9668: 9664: 9660: 9651: 9648: 9645: 9640: 9634: 9630: 9626: 9620: 9615: 9610: 9606: 9600: 9597: 9594: 9591: 9589: 9584: 9581: 9578: 9575: 9571: 9564: 9561: 9558: 9555: 9552: 9549: 9544: 9540: 9536: 9533: 9530: 9525: 9521: 9515: 9512: 9509: 9506: 9503: 9498: 9494: 9490: 9487: 9484: 9476: 9471: 9468: 9465: 9461: 9457: 9454: 9452: 9450: 9447: 9444: 9441: 9440: 9435: 9432: 9427: 9423: 9419: 9415: 9406: 9403: 9400: 9395: 9389: 9385: 9381: 9375: 9370: 9364: 9360: 9356: 9350: 9345: 9340: 9336: 9330: 9327: 9324: 9321: 9319: 9314: 9311: 9308: 9305: 9301: 9294: 9291: 9288: 9285: 9282: 9276: 9272: 9269: 9264: 9260: 9255: 9249: 9245: 9239: 9236: 9232: 9223: 9218: 9215: 9212: 9208: 9204: 9201: 9199: 9197: 9194: 9191: 9188: 9187: 9184: 9176: 9173: 9170: 9164: 9161: 9155: 9151: 9145: 9139: 9136: 9130: 9126: 9120: 9117: 9114: 9111: 9109: 9103: 9100: 9097: 9094: 9091: 9085: 9082: 9078: 9070: 9065: 9062: 9059: 9055: 9051: 9048: 9046: 9044: 9041: 9038: 9035: 9034: 9031: 9023: 9020: 9017: 9011: 9008: 9002: 8998: 8992: 8986: 8983: 8977: 8973: 8967: 8964: 8961: 8958: 8956: 8950: 8947: 8944: 8941: 8938: 8935: 8932: 8926: 8923: 8920: 8917: 8913: 8905: 8900: 8897: 8894: 8890: 8886: 8883: 8881: 8879: 8876: 8873: 8870: 8869: 8851: 8848: 8829: 8808: 8804:. The numbers 8783: 8780: 8777: 8774: 8768: 8765: 8762: 8758: 8754: 8750: 8741: 8738: 8735: 8730: 8725: 8721: 8715: 8710: 8705: 8701: 8695: 8692: 8689: 8686: 8684: 8679: 8676: 8673: 8670: 8666: 8659: 8656: 8653: 8650: 8643: 8639: 8635: 8632: 8629: 8621: 8616: 8613: 8610: 8606: 8602: 8599: 8597: 8595: 8592: 8589: 8586: 8585: 8582: 8579: 8575: 8571: 8567: 8558: 8555: 8552: 8547: 8541: 8537: 8533: 8527: 8522: 8517: 8513: 8507: 8504: 8501: 8496: 8493: 8488: 8485: 8483: 8478: 8475: 8472: 8469: 8465: 8458: 8455: 8452: 8449: 8446: 8443: 8438: 8434: 8430: 8427: 8424: 8419: 8415: 8409: 8406: 8403: 8400: 8397: 8389: 8384: 8381: 8378: 8374: 8370: 8365: 8362: 8357: 8354: 8352: 8350: 8347: 8344: 8341: 8338: 8333: 8330: 8325: 8322: 8320: 8318: 8315: 8312: 8309: 8308: 8305: 8302: 8298: 8294: 8290: 8281: 8278: 8275: 8270: 8264: 8260: 8256: 8250: 8245: 8240: 8236: 8230: 8227: 8224: 8221: 8219: 8214: 8211: 8208: 8205: 8201: 8194: 8191: 8188: 8185: 8182: 8179: 8174: 8170: 8166: 8163: 8160: 8155: 8151: 8145: 8142: 8139: 8136: 8133: 8125: 8120: 8117: 8114: 8110: 8106: 8103: 8101: 8099: 8096: 8093: 8090: 8089: 8084: 8081: 8076: 8072: 8068: 8064: 8055: 8052: 8049: 8044: 8038: 8034: 8030: 8024: 8019: 8014: 8010: 8004: 8001: 7998: 7995: 7993: 7988: 7985: 7981: 7974: 7971: 7968: 7965: 7962: 7955: 7952: 7948: 7942: 7938: 7934: 7931: 7928: 7920: 7915: 7912: 7909: 7905: 7901: 7898: 7896: 7894: 7891: 7888: 7885: 7884: 7879: 7876: 7871: 7867: 7863: 7859: 7850: 7847: 7844: 7839: 7833: 7829: 7825: 7819: 7814: 7809: 7805: 7799: 7796: 7793: 7790: 7788: 7783: 7780: 7777: 7774: 7770: 7763: 7760: 7757: 7754: 7751: 7745: 7739: 7735: 7731: 7728: 7724: 7718: 7714: 7710: 7707: 7704: 7699: 7696: 7692: 7683: 7678: 7675: 7672: 7668: 7664: 7661: 7659: 7657: 7654: 7651: 7648: 7647: 7644: 7636: 7633: 7630: 7624: 7621: 7615: 7611: 7605: 7599: 7596: 7590: 7586: 7580: 7577: 7574: 7571: 7569: 7564: 7561: 7557: 7550: 7547: 7544: 7541: 7538: 7531: 7527: 7523: 7520: 7517: 7509: 7504: 7501: 7498: 7494: 7490: 7487: 7485: 7483: 7480: 7477: 7474: 7473: 7470: 7462: 7459: 7456: 7450: 7447: 7441: 7437: 7431: 7425: 7422: 7416: 7412: 7406: 7403: 7400: 7397: 7395: 7390: 7387: 7384: 7381: 7377: 7370: 7367: 7364: 7361: 7358: 7355: 7352: 7345: 7341: 7337: 7334: 7331: 7323: 7318: 7315: 7312: 7308: 7304: 7301: 7299: 7297: 7294: 7291: 7288: 7287: 7269: 7266: 7256:When only the 7239: 7234: 7230: 7221: 7217: 7213: 7210: 7207: 7202: 7198: 7192: 7189: 7186: 7183: 7180: 7175: 7171: 7167: 7164: 7161: 7153: 7148: 7145: 7142: 7138: 7134: 7131: 7129: 7126: 7121: 7117: 7111: 7108: 7103: 7098: 7094: 7088: 7085: 7080: 7075: 7071: 7065: 7062: 7057: 7052: 7048: 7042: 7039: 7034: 7031: 7026: 7023: 7018: 7015: 7012: 7009: 7007: 7001: 6998: 6993: 6989: 6985: 6982: 6979: 6976: 6973: 6972: 6969: 6964: 6960: 6953: 6950: 6947: 6944: 6941: 6938: 6933: 6929: 6925: 6922: 6919: 6914: 6910: 6904: 6901: 6898: 6895: 6892: 6887: 6884: 6881: 6877: 6873: 6870: 6867: 6859: 6854: 6851: 6848: 6844: 6840: 6837: 6835: 6832: 6827: 6823: 6817: 6814: 6809: 6804: 6800: 6794: 6791: 6786: 6781: 6777: 6771: 6768: 6763: 6758: 6754: 6748: 6745: 6740: 6737: 6732: 6729: 6724: 6721: 6718: 6715: 6713: 6708: 6705: 6700: 6696: 6693: 6690: 6687: 6684: 6683: 6595: 6592: 6588: 6584: 6580: 6546: 6540: 6537: 6532: 6529: 6526: 6523: 6520: 6517: 6514: 6511: 6508: 6505: 6502: 6499: 6496: 6493: 6487: 6482: 6478: 6475: 6472: 6469: 6466: 6458: 6453: 6450: 6447: 6443: 6439: 6433: 6428: 6425: 6420: 6388: 6384: 6377: 6372: 6369: 6364: 6356: 6351: 6348: 6345: 6341: 6337: 6332: 6328: 6324: 6321: 6318: 6315: 6297: 6294: 6277: 6274: 6270: 6266: 6262: 6235: 6230: 6227: 6224: 6220: 6214: 6210: 6207: 6204: 6201: 6198: 6195: 6187: 6182: 6179: 6176: 6172: 6168: 6165: 6163: 6156: 6152: 6148: 6145: 6142: 6139: 6135: 6130: 6129: 6124: 6121: 6118: 6114: 6110: 6105: 6100: 6097: 6094: 6090: 6086: 6083: 6081: 6074: 6070: 6066: 6063: 6060: 6057: 6053: 6048: 6047: 6042: 6038: 6032: 6027: 6024: 6021: 6017: 6013: 6010: 6008: 6003: 6000: 5997: 5993: 5988: 5987: 5969: 5966: 5957:converges for 5942:converges for 5923: 5920: 5916: 5912: 5908: 5890:, named after 5869: 5866: 5863: 5858: 5853: 5849: 5843: 5838: 5833: 5829: 5823: 5820: 5817: 5812: 5807: 5803: 5795: 5792: 5789: 5785: 5781: 5778: 5775: 5770: 5765: 5762: 5759: 5755: 5751: 5748: 5746: 5744: 5741: 5738: 5735: 5732: 5729: 5726: 5723: 5722: 5719: 5716: 5713: 5708: 5703: 5699: 5693: 5688: 5683: 5679: 5673: 5670: 5667: 5664: 5659: 5654: 5650: 5642: 5637: 5634: 5631: 5627: 5623: 5620: 5617: 5615: 5613: 5610: 5607: 5604: 5601: 5598: 5595: 5592: 5591: 5567: 5564: 5550: 5546: 5539: 5536: 5530: 5526: 5518: 5513: 5510: 5507: 5503: 5499: 5496: 5493: 5490: 5486: 5482: 5479: 5476: 5473: 5470: 5433: 5430: 5427: 5421: 5418: 5412: 5408: 5402: 5396: 5393: 5387: 5383: 5377: 5374: 5371: 5368: 5365: 5359: 5356: 5350: 5346: 5338: 5333: 5330: 5327: 5323: 5319: 5314: 5310: 5279: 5275: 5235: 5232: 5217: 5214: 5195: 5192: 5189: 5186: 5181: 5178: 5175: 5171: 5167: 5163: 5159: 5156: 5153: 5150: 5147: 5144: 5141: 5136: 5131: 5128: 5124: 5116: 5112: 5108: 5105: 5101: 5097: 5094: 5091: 5088: 5085: 5082: 5079: 5001:expected value 4986: 4980: 4977: 4970: 4966: 4962: 4958: 4954: 4951: 4945: 4942: 4939: 4936: 4933: 4930: 4927: 4922: 4917: 4914: 4911: 4907: 4901: 4897: 4893: 4890: 4886: 4878: 4874: 4870: 4867: 4863: 4859: 4856: 4853: 4850: 4847: 4844: 4841: 4817: 4812: 4809: 4806: 4802: 4795: 4792: 4786: 4782: 4774: 4769: 4766: 4763: 4759: 4753: 4750: 4746: 4742: 4737: 4733: 4727: 4723: 4716: 4713: 4707: 4703: 4695: 4690: 4687: 4684: 4680: 4662: 4626: 4607: 4604: 4601: 4598: 4595: 4592: 4589: 4586: 4579: 4575: 4570: 4567: 4564: 4561: 4556: 4551: 4547: 4537: 4534: 4528: 4524: 4516: 4511: 4508: 4505: 4501: 4493: 4489: 4485: 4482: 4478: 4437: 4436:Generalization 4434: 4416:Laurent series 4256: 4251: 4248: 4245: 4237: 4235: 4232: 4231: 4228: 4225: 4222: 4214: 4208: 4204: 4199: 4195: 4192: 4188: 4184: 4183: 4181: 4176: 4173: 4170: 4167: 4164: 4104: 4008: 4002: 3999: 3993: 3989: 3983: 3977: 3974: 3968: 3964: 3958: 3952: 3949: 3943: 3939: 3933: 3930: 3927: 3923: 3919: 3916: 3823:Main article: 3820: 3817: 3816: 3815: 3812: 3801: 3794:Chebyshev form 3790: 3771: 3768: 3654: 3650: 3646: 3640: 3637: 3632: 3629: 3626: 3621: 3618: 3615: 3611: 3565: 3560: 3556: 3552: 3549: 3546: 3543: 3538: 3534: 3528: 3523: 3520: 3517: 3513: 3509: 3506: 3503: 3500: 3497: 3430:Main article: 3427: 3424: 3386: 3380: 3377: 3371: 3366: 3362: 3358: 3355: 3352: 3332: 3327: 3320: 3316: 3310: 3303: 3298: 3295: 3282:, the inverse 3261: 3256: 3253: 3250: 3244: 3241: 3233: 3224: 3221: 3215: 3212: 3208: 3187: 3166: 3163: 3160: 3156: 3137: 3134: 3131: 3128: 3109: 3106: 3103: 3100: 3081: 3078: 3075: 3072: 3046: 3043: 3040: 3037: 3016: 3013: 3010: 3007: 2988: 2985: 2982: 2979: 2959: 2956: 2953: 2950: 2931:In late 1670, 2926:Madhava series 2875:Zeno's paradox 2863: 2860: 2804: 2801: 2798: 2793: 2788: 2784: 2778: 2773: 2768: 2764: 2758: 2753: 2748: 2744: 2738: 2733: 2728: 2724: 2718: 2715: 2712: 2709: 2706: 2703: 2701: 2699: 2696: 2693: 2687: 2684: 2678: 2674: 2668: 2662: 2659: 2653: 2649: 2643: 2637: 2634: 2628: 2624: 2618: 2612: 2609: 2603: 2599: 2593: 2587: 2584: 2578: 2574: 2568: 2562: 2559: 2553: 2549: 2543: 2540: 2538: 2533: 2530: 2524: 2520: 2512: 2507: 2504: 2501: 2497: 2493: 2492: 2459: 2456: 2453: 2448: 2443: 2438: 2434: 2431: 2428: 2424: 2413: 2409: 2405: 2400: 2397: 2394: 2391: 2388: 2385: 2380: 2377: 2372: 2369: 2366: 2363: 2328: 2325: 2322: 2317: 2313: 2309: 2306: 2303: 2300: 2294: 2291: 2285: 2280: 2276: 2272: 2269: 2266: 2263: 2257: 2254: 2248: 2243: 2239: 2235: 2232: 2229: 2226: 2220: 2217: 2211: 2208: 2205: 2202: 2199: 2196: 2158: 2155: 2152: 2147: 2143: 2136: 2133: 2127: 2122: 2118: 2111: 2108: 2102: 2097: 2093: 2086: 2083: 2077: 2074: 2071: 2031: 2028: 2025: 2020: 2016: 2012: 2009: 2006: 2003: 2000: 1995: 1991: 1987: 1984: 1981: 1978: 1975: 1972: 1969: 1966: 1963: 1960: 1957: 1954: 1892: 1889: 1886: 1881: 1877: 1873: 1868: 1864: 1860: 1857: 1854: 1851: 1806: 1803: 1791: 1786: 1782: 1775: 1772: 1767: 1764: 1761: 1756: 1753: 1750: 1746: 1737: 1732: 1729: 1726: 1722: 1718: 1715: 1712: 1707: 1703: 1696: 1693: 1688: 1685: 1682: 1678: 1675: 1668: 1663: 1659: 1652: 1649: 1644: 1641: 1638: 1634: 1631: 1624: 1621: 1615: 1612: 1607: 1604: 1601: 1597: 1594: 1587: 1584: 1581: 1578: 1575: 1557:sigma notation 1475: 1470: 1466: 1462: 1459: 1456: 1453: 1447: 1444: 1439: 1436: 1433: 1428: 1425: 1422: 1418: 1409: 1404: 1401: 1398: 1394: 1390: 1387: 1384: 1379: 1375: 1371: 1368: 1365: 1362: 1356: 1353: 1348: 1345: 1342: 1338: 1335: 1328: 1323: 1319: 1315: 1312: 1309: 1306: 1300: 1297: 1292: 1289: 1286: 1282: 1279: 1272: 1269: 1266: 1263: 1260: 1257: 1251: 1248: 1243: 1240: 1237: 1233: 1230: 1223: 1220: 1217: 1214: 1211: 1192:complex number 1161: 1158: 1042: 1041: 1039: 1038: 1031: 1024: 1016: 1013: 1012: 1009: 1008: 1003: 998: 993: 991:List of topics 988: 983: 978: 972: 967: 966: 963: 962: 959: 958: 953: 948: 943: 937: 932: 931: 928: 927: 922: 921: 920: 919: 914: 909: 899: 894: 893: 890: 889: 884: 883: 882: 881: 876: 871: 866: 861: 856: 851: 843: 842: 838: 837: 836: 835: 830: 825: 820: 812: 811: 805: 798: 797: 794: 793: 788: 787: 786: 785: 780: 775: 770: 765: 760: 752: 751: 747: 746: 745: 744: 739: 734: 729: 724: 719: 709: 702: 701: 698: 697: 692: 691: 690: 689: 684: 679: 674: 669: 663: 658: 653: 648: 643: 635: 634: 628: 627: 626: 625: 620: 615: 610: 605: 600: 585: 578: 577: 574: 573: 568: 567: 566: 565: 560: 555: 550: 548:Changing order 545: 535: 530: 512: 507: 502: 494: 493: 492:Integration by 489: 488: 487: 486: 481: 476: 471: 466: 456: 454:Antiderivative 448: 447: 443: 442: 441: 440: 435: 430: 420: 413: 412: 409: 408: 403: 402: 401: 400: 395: 390: 385: 380: 375: 370: 365: 360: 355: 347: 346: 340: 339: 338: 337: 332: 327: 322: 317: 312: 304: 303: 299: 298: 297: 296: 295: 294: 289: 284: 274: 261: 260: 254: 247: 246: 243: 242: 240: 239: 234: 229: 223: 221: 220: 215: 209: 208: 207: 199: 198: 186: 183: 180: 177: 174: 171: 168: 165: 162: 159: 156: 153: 149: 146: 143: 139: 136: 130: 125: 121: 111: 108: 107: 101: 100: 26: 9: 6: 4: 3: 2: 20076: 20065: 20062: 20060: 20057: 20055: 20054:Real analysis 20052: 20051: 20049: 20034: 20026: 20025: 20022: 20016: 20013: 20011: 20008: 20006: 20003: 20001: 19998: 19996: 19993: 19991: 19988: 19987: 19985: 19983: 19979: 19973: 19970: 19968: 19965: 19963: 19960: 19958: 19955: 19953: 19950: 19948: 19945: 19943: 19940: 19938: 19935: 19933: 19932:Taylor series 19930: 19929: 19927: 19923: 19913: 19910: 19908: 19905: 19903: 19900: 19898: 19895: 19893: 19890: 19888: 19885: 19883: 19880: 19878: 19875: 19873: 19870: 19868: 19865: 19864: 19862: 19858: 19852: 19849: 19847: 19844: 19842: 19839: 19837: 19834: 19833: 19831: 19827: 19824: 19820: 19810: 19807: 19805: 19802: 19800: 19797: 19796: 19794: 19790: 19784: 19781: 19779: 19776: 19774: 19771: 19769: 19766: 19765: 19763: 19759: 19756: 19752: 19746: 19743: 19741: 19738: 19736: 19733: 19732: 19730: 19726: 19721: 19705: 19702: 19701: 19700: 19697: 19695: 19692: 19690: 19687: 19685: 19682: 19680: 19677: 19675: 19672: 19670: 19667: 19665: 19662: 19660: 19657: 19655: 19652: 19651: 19649: 19645: 19638: 19632: 19629: 19627: 19624: 19622: 19621:Powers of two 19619: 19617: 19614: 19612: 19609: 19607: 19606:Square number 19604: 19602: 19599: 19597: 19594: 19592: 19589: 19588: 19586: 19582: 19579: 19577: 19573: 19569: 19565: 19558: 19553: 19551: 19546: 19544: 19539: 19538: 19535: 19526: 19525: 19520: 19517: 19512: 19508: 19504: 19503: 19498: 19494: 19493: 19482: 19477: 19473: 19471:0-201-53174-7 19467: 19463: 19458: 19454: 19450: 19449: 19443: 19439: 19433: 19429: 19428: 19422: 19418: 19416:0-07-099557-5 19412: 19408: 19404: 19403:Rudin, Walter 19400: 19396: 19391: 19381:on 2023-03-14 19377: 19373: 19369: 19365: 19361: 19354: 19345: 19341: 19337: 19333: 19329: 19325: 19321: 19318:(2): 97–137. 19317: 19313: 19308: 19304: 19298: 19294: 19289: 19285: 19279: 19275: 19271: 19267: 19263: 19258: 19254: 19252:0-19-506135-7 19248: 19243: 19242: 19235: 19231: 19225: 19221: 19217: 19213: 19209: 19205: 19201: 19197: 19193: 19189: 19185: 19181: 19177: 19172: 19168: 19164: 19160: 19156: 19152: 19150:0-13-321431-1 19146: 19141: 19140: 19133: 19129: 19125: 19121: 19119:9789971512989 19115: 19111: 19107: 19103: 19099: 19095: 19091: 19087: 19083: 19079: 19074: 19070: 19066: 19062: 19058: 19054: 19050: 19045: 19041: 19037: 19032: 19028: 19026:0-471-09763-2 19022: 19017: 19016: 19010: 19005: 19001: 18995: 18991: 18986: 18982: 18976: 18972: 18971: 18965: 18961: 18957: 18956: 18951: 18947: 18943: 18942: 18927: 18924: 18921: 18918: 18917: 18913: 18906: 18902: 18897: 18895: 18887: 18883: 18878: 18871: 18867: 18863: 18858: 18851: 18847: 18842: 18835: 18831: 18826: 18819: 18814: 18805: 18801: 18798: 18796: 18792: 18789: 18788: 18784: 18782: 18774: 18770: 18765: 18758: 18753: 18743: 18740: 18737: 18734: 18733: 18729: 18722: 18717: 18708: 18705: 18702: 18698: 18694: 18691: 18690: 18686: 18677: 18674: 18672: 18669: 18666: 18665:Turnbull 1939 18663: 18662: 18658: 18651: 18646: 18639: 18635: 18630: 18623: 18619: 18614: 18608:, p. 33. 18607: 18606:Lindberg 2007 18602: 18595: 18590: 18583: 18579: 18574: 18570: 18561: 18558: 18556: 18553: 18550: 18547: 18544: 18541: 18539: 18536: 18534: 18531: 18530: 18526: 18520: 18515: 18504: 18501: 18497: 18493: 18489: 18484: 18481: 18475: 18471: 18466: 18462: 18459: 18455: 18454:neighbourhood 18451: 18447: 18441: 18435: 18431: 18424: 18420: 18415: 18414: 18413: 18411: 18407: 18403: 18399: 18395: 18391: 18385: 18375: 18362: 18359: 18351: 18342: 18339: 18336: 18331: 18327: 18320: 18317: 18311: 18308: 18305: 18302: 18299: 18296: 18290: 18287: 18284: 18278: 18275: 18270: 18266: 18257: 18254:| < 1 18252: 18243: 18236: 18219: 18216: 18211: 18207: 18200: 18197: 18191: 18188: 18185: 18182: 18179: 18176: 18174: 18166: 18163: 18151: 18143: 18140: 18137: 18128: 18125: 18119: 18113: 18110: 18107: 18098: 18095: 18092: 18086: 18083: 18078: 18070: 18067: 18064: 18058: 18046: 18043: 18038: 18032: 18029: 18026: 18020: 18017: 18011: 18008: 18005: 17999: 17996: 17993: 17990: 17988: 17980: 17977: 17974: 17968: 17956: 17953: 17936: 17933: 17929: 17922: 17919: 17916: 17908: 17905: 17901: 17895: 17887: 17884: 17881: 17875: 17869: 17866: 17863: 17855: 17852: 17848: 17841: 17838: 17835: 17826: 17823: 17820: 17814: 17811: 17805: 17802: 17799: 17791: 17788: 17784: 17778: 17770: 17767: 17764: 17757: 17750: 17747: 17743: 17738: 17734: 17723: 17720: 17717: 17709: 17705: 17698: 17695: 17692: 17686: 17680: 17677: 17674: 17666: 17662: 17655: 17652: 17649: 17643: 17637: 17634: 17631: 17625: 17620: 17614: 17611: 17608: 17602: 17590: 17587: 17570: 17567: 17561: 17558: 17555: 17547: 17544: 17540: 17536: 17534: 17526: 17523: 17520: 17512: 17509: 17505: 17497: 17494: 17491: 17489: 17481: 17478: 17475: 17467: 17464: 17460: 17452: 17449: 17447: 17439: 17436: 17433: 17425: 17422: 17418: 17410: 17407: 17405: 17397: 17394: 17391: 17383: 17379: 17371: 17368: 17366: 17358: 17355: 17352: 17344: 17340: 17327: 17324: 17307: 17301: 17298: 17295: 17289: 17285: 17279: 17274: 17271: 17267: 17263: 17261: 17254: 17251: 17247: 17234: 17226: 17223: 17220: 17211: 17207: 17201: 17198: 17196: 17189: 17186: 17182: 17171: 17168: 17165: 17159: 17156: 17151: 17147: 17143: 17141: 17134: 17131: 17127: 17116: 17113: 17110: 17104: 17100: 17094: 17092: 17085: 17081: 17070: 17067: 17064: 17058: 17055: 17050: 17046: 17042: 17040: 17033: 17029: 17016: 17013: 17000: 16994: 16991: 16988: 16982: 16979: 16974: 16970: 16966: 16960: 16957: 16954: 16948: 16938: 16934: 16922: 16918: 16914: 16910: 16906: 16900: 16891: 16889: 16884: 16871: 16856: 16852: 16846: 16834: 16827: 16824: 16817: 16804: 16788: 16785: 16777: 16768: 16764: 16750: 16742: 16740: 16736: 16730: 16726: 16723: 16717: 16713: 16709:evaluated at 16704: 16698: 16694: 16691: 16685: 16672: 16669: 16666: 16655: 16643: 16628: 16623: 16619: 16614: 16593: 16579: 16576: 16572: 16567: 16553: 16550: 16530: 16519: 16505: 16502: 16488: 16480: 16476: 16466: 16464: 16459: 16436: 16433: 16430: 16422: 16419: 16415: 16409: 16401: 16398: 16395: 16389: 16383: 16380: 16377: 16369: 16366: 16362: 16355: 16352: 16349: 16340: 16337: 16334: 16328: 16325: 16319: 16316: 16313: 16305: 16302: 16298: 16292: 16284: 16281: 16278: 16262: 16259: 16255: 16250: 16244: 16241: 16238: 16230: 16226: 16219: 16216: 16213: 16207: 16201: 16198: 16195: 16187: 16183: 16176: 16173: 16170: 16164: 16158: 16155: 16152: 16146: 16138: 16134: 16130: 16101: 16098: 16095: 16089: 16080: 16063: 16060: 16052: 16048: 16044: 16039: 16035: 16023: 16019: 16015: 16010: 16006: 15994: 15990: 15986: 15981: 15977: 15965: 15961: 15952: 15948: 15939: 15935: 15921: 15917: 15913: 15910: 15907: 15902: 15898: 15891: 15886: 15873: 15868: 15865: 15862: 15858: 15852: 15847: 15844: 15841: 15837: 15831: 15826: 15823: 15820: 15816: 15809: 15806: 15802: 15797: 15793: 15780: 15776: 15772: 15767: 15763: 15751: 15747: 15743: 15738: 15734: 15722: 15718: 15709: 15705: 15691: 15687: 15683: 15680: 15677: 15672: 15668: 15661: 15656: 15643: 15638: 15635: 15632: 15628: 15622: 15617: 15614: 15611: 15607: 15600: 15597: 15593: 15588: 15580: 15576: 15572: 15567: 15563: 15551: 15547: 15533: 15529: 15525: 15522: 15519: 15514: 15510: 15503: 15492: 15487: 15484: 15481: 15477: 15473: 15465: 15461: 15457: 15454: 15451: 15446: 15442: 15435: 15432: 15430: 15417: 15413: 15409: 15406: 15403: 15398: 15394: 15386: 15376: 15372: 15366: 15362: 15355: 15348: 15344: 15338: 15334: 15325: 15318: 15314: 15310: 15307: 15304: 15299: 15295: 15283: 15275: 15270: 15266: 15262: 15259: 15254: 15250: 15240: 15236: 15225: 15221: 15217: 15212: 15208: 15201: 15194: 15190: 15179: 15175: 15171: 15166: 15162: 15145: 15142: 15137: 15133: 15128: 15124: 15114: 15111: 15106: 15102: 15097: 15093: 15091: 15081: 15077: 15073: 15070: 15067: 15062: 15058: 15051: 15039: 15031: 15029: 15025: 15020: 15018: 15014: 15010: 15005: 15003: 14999: 14995: 14991: 14987: 14984:Classically, 14977: 14960: 14955: 14951: 14944: 14941: 14936: 14933: 14930: 14917: 14914: 14911: 14907: 14903: 14901: 14891: 14887: 14880: 14877: 14872: 14869: 14866: 14853: 14850: 14847: 14843: 14839: 14836: 14833: 14831: 14821: 14817: 14812: 14805: 14799: 14796: 14793: 14786: 14781: 14775: 14772: 14768: 14762: 14751: 14748: 14745: 14741: 14737: 14734: 14731: 14725: 14719: 14716: 14713: 14704: 14700: 14687: 14684: 14681: 14677: 14673: 14667: 14664: 14658: 14654: 14641: 14638: 14635: 14631: 14627: 14624: 14621: 14619: 14608: 14605: 14599: 14596: 14593: 14589: 14576: 14573: 14570: 14566: 14562: 14556: 14553: 14547: 14543: 14530: 14527: 14524: 14520: 14516: 14513: 14510: 14504: 14501: 14495: 14492: 14489: 14485: 14472: 14469: 14466: 14462: 14458: 14452: 14449: 14443: 14439: 14426: 14423: 14420: 14416: 14412: 14407: 14403: 14399: 14396: 14391: 14387: 14383: 14381: 14374: 14370: 14363: 14360: 14357: 14342: 14339: 14326: 14323: 14320: 14314: 14311: 14305: 14301: 14295: 14289: 14286: 14280: 14276: 14270: 14264: 14261: 14255: 14251: 14245: 14242: 14239: 14236: 14233: 14227: 14224: 14218: 14214: 14201: 14198: 14195: 14191: 14187: 14182: 14178: 14169: 14166: 14156: 14152: 14143:Third example 14140: 14127: 14124: 14121: 14116: 14112: 14105: 14102: 14096: 14091: 14087: 14080: 14077: 14071: 14066: 14062: 14058: 14055: 14052: 14049: 14046: 14040: 14037: 14034: 14028: 14024: 14013: 13998: 13995: 13992: 13969: 13965: 13941: 13935: 13913: 13909: 13899: 13886: 13883: 13877: 13871: 13868: 13862: 13857: 13853: 13846: 13843: 13837: 13832: 13828: 13821: 13818: 13812: 13807: 13803: 13793: 13787: 13784: 13778: 13773: 13769: 13762: 13759: 13753: 13748: 13744: 13734: 13728: 13725: 13719: 13714: 13710: 13703: 13700: 13694: 13689: 13685: 13675: 13672: 13669: 13664: 13660: 13650: 13647: 13644: 13639: 13635: 13626: 13613: 13608: 13604: 13583: 13580: 13577: 13571: 13565: 13556: 13539: 13536: 13531: 13527: 13522: 13515: 13512: 13506: 13502: 13496: 13491: 13486: 13482: 13476: 13471: 13467: 13462: 13458: 13453: 13449: 13444: 13438: 13433: 13429: 13423: 13418: 13414: 13409: 13405: 13400: 13396: 13391: 13385: 13380: 13376: 13370: 13365: 13361: 13356: 13352: 13349: 13344: 13340: 13336: 13331: 13327: 13323: 13321: 13312: 13308: 13305: 13299: 13296: 13290: 13286: 13280: 13274: 13271: 13265: 13261: 13255: 13252: 13248: 13243: 13239: 13236: 13231: 13227: 13221: 13217: 13213: 13208: 13204: 13198: 13194: 13190: 13185: 13181: 13175: 13171: 13167: 13164: 13159: 13155: 13151: 13146: 13142: 13137: 13133: 13131: 13124: 13120: 13107: 13093: 13090: 13087: 13078: 13065: 13062: 13057: 13053: 13047: 13043: 13039: 13034: 13030: 13024: 13020: 13016: 13011: 13007: 13001: 12997: 12993: 12990: 12985: 12981: 12977: 12972: 12968: 12964: 12958: 12955: 12952: 12946: 12942: 12931: 12928: 12915: 12912: 12909: 12903: 12900: 12894: 12890: 12884: 12878: 12875: 12869: 12865: 12859: 12856: 12853: 12850: 12847: 12844: 12836: 12833: 12820: 12817: 12814: 12808: 12805: 12799: 12795: 12789: 12783: 12780: 12774: 12770: 12764: 12758: 12755: 12749: 12745: 12739: 12736: 12733: 12730: 12727: 12722: 12718: 12709: 12706: 12692: 12686: 12683: 12680: 12674: 12670: 12664: 12658: 12652: 12644: 12636: 12634: 12633:even function 12629: 12611: 12606: 12601: 12597: 12593: 12588: 12585: 12580: 12575: 12571: 12565: 12560: 12555: 12551: 12545: 12540: 12535: 12531: 12525: 12522: 12520: 12510: 12504: 12496: 12493: 12490: 12487: 12484: 12477: 12472: 12469: 12464: 12456: 12453: 12450: 12447: 12444: 12435: 12432: 12426: 12421: 12413: 12410: 12407: 12404: 12401: 12392: 12389: 12383: 12377: 12374: 12371: 12368: 12365: 12359: 12357: 12339: 12336: 12333: 12330: 12327: 12321: 12318: 12308: 12305: 12302: 12300: 12292: 12286: 12274: 12270: 12257: 12252: 12247: 12243: 12239: 12234: 12231: 12226: 12221: 12217: 12211: 12206: 12201: 12197: 12191: 12186: 12181: 12177: 12171: 12168: 12165: 12162: 12159: 12156: 12153: 12145: 12142: 12127: 12122: 12118: 12114: 12109: 12106: 12101: 12096: 12092: 12086: 12081: 12076: 12072: 12066: 12063: 12060: 12054: 12051: 12048: 12042: 12039: 12031: 12029: 12013: 12010: 12007: 12004: 12001: 11995: 11972: 11969: 11966: 11960: 11957: 11951: 11942: 11929: 11916: 11913: 11910: 11907: 11904: 11898: 11895: 11884: 11880: 11874: 11868: 11860: 11857: 11844: 11831: 11828: 11822: 11815: 11812: 11806: 11795: 11792: 11788: 11782: 11779: 11776: 11770: 11767: 11764: 11758: 11752: 11744: 11739:First example 11736: 11734: 11730: 11719: 11701: 11698: 11695: 11692: 11688: 11684: 11681: 11677: 11665: 11662: 11659: 11655: 11651: 11646: 11642: 11635: 11629: 11619: 11616: 11613: 11609: 11605: 11600: 11596: 11592: 11577: 11573: 11566: 11558: 11550: 11546: 11542: 11537: 11529: 11521: 11517: 11501: 11497: 11493: 11490: 11482: 11474: 11470: 11464: 11460: 11456: 11448: 11440: 11436: 11427: 11424: 11406: 11402: 11398: 11395: 11391: 11379: 11376: 11373: 11369: 11365: 11360: 11356: 11349: 11343: 11333: 11330: 11327: 11323: 11319: 11314: 11310: 11306: 11303: 11288: 11284: 11277: 11269: 11261: 11257: 11253: 11248: 11240: 11232: 11228: 11212: 11208: 11204: 11201: 11193: 11185: 11181: 11175: 11171: 11167: 11164: 11156: 11148: 11144: 11135: 11133: 11128: 11107: 11103: 11098: 11092: 11084: 11081: 11068: 11065: 11062: 11058: 11054: 11051: 11048: 11045: 11043: 11035: 11027: 11023: 11011: 11007: 11002: 10991: 10988: 10985: 10981: 10977: 10974: 10971: 10968: 10966: 10958: 10950: 10946: 10933: 10931: 10926: 10907: 10904: 10900: 10891: 10883: 10880: 10872: 10868: 10861: 10858: 10855: 10852: 10842: 10834: 10828: 10825: 10806: 10803: 10800: 10796: 10792: 10790: 10782: 10776: 10771: 10768: 10757: 10754: 10750: 10741: 10733: 10730: 10722: 10718: 10710: 10702: 10696: 10693: 10674: 10671: 10668: 10664: 10660: 10658: 10650: 10644: 10639: 10636: 10622: 10620: 10617:The complete 10610: 10608: 10603: 10584: 10581: 10578: 10575: 10571: 10562: 10554: 10551: 10548: 10545: 10535: 10527: 10524: 10508: 10505: 10502: 10498: 10494: 10492: 10484: 10476: 10460: 10457: 10454: 10451: 10447: 10438: 10430: 10427: 10424: 10421: 10411: 10403: 10400: 10384: 10381: 10378: 10374: 10370: 10368: 10360: 10352: 10333: 10331: 10330: 10324: 10305: 10302: 10299: 10296: 10292: 10283: 10275: 10272: 10269: 10266: 10259: 10247: 10244: 10241: 10237: 10233: 10231: 10223: 10215: 10211: 10201: 10198: 10195: 10192: 10188: 10179: 10171: 10168: 10165: 10162: 10155: 10143: 10140: 10137: 10133: 10129: 10127: 10119: 10111: 10107: 10094: 10092: 10087: 10068: 10064: 10056: 10052: 10048: 10036: 10033: 10030: 10026: 10022: 10020: 10012: 10004: 9988: 9984: 9976: 9972: 9968: 9956: 9953: 9950: 9946: 9942: 9940: 9932: 9924: 9905: 9903: 9893: 9891: 9886: 9879: 9870: 9853: 9850: 9847: 9844: 9838: 9835: 9832: 9824: 9808: 9805: 9800: 9795: 9791: 9785: 9780: 9775: 9771: 9765: 9762: 9759: 9757: 9748: 9745: 9742: 9739: 9733: 9730: 9727: 9724: 9720: 9707: 9704: 9701: 9697: 9693: 9691: 9686: 9683: 9680: 9673: 9670: 9662: 9646: 9643: 9638: 9632: 9628: 9624: 9618: 9613: 9608: 9604: 9598: 9595: 9592: 9590: 9582: 9579: 9576: 9573: 9569: 9559: 9556: 9553: 9550: 9542: 9534: 9531: 9523: 9519: 9513: 9507: 9504: 9496: 9488: 9485: 9469: 9466: 9463: 9459: 9455: 9453: 9448: 9445: 9442: 9433: 9430: 9425: 9417: 9401: 9398: 9393: 9387: 9383: 9379: 9373: 9368: 9362: 9358: 9354: 9348: 9343: 9338: 9334: 9328: 9325: 9322: 9320: 9312: 9309: 9306: 9303: 9299: 9292: 9286: 9283: 9274: 9270: 9267: 9262: 9258: 9253: 9247: 9243: 9237: 9234: 9230: 9216: 9213: 9210: 9206: 9202: 9200: 9195: 9192: 9189: 9182: 9179:for all 9171: 9168: 9162: 9159: 9153: 9149: 9143: 9137: 9134: 9128: 9124: 9118: 9115: 9112: 9110: 9101: 9095: 9092: 9083: 9080: 9076: 9063: 9060: 9057: 9053: 9049: 9047: 9042: 9039: 9036: 9029: 9026:for all 9018: 9015: 9009: 9006: 9000: 8996: 8990: 8984: 8981: 8975: 8971: 8965: 8962: 8959: 8957: 8948: 8942: 8939: 8936: 8933: 8924: 8921: 8918: 8915: 8911: 8898: 8895: 8892: 8888: 8884: 8882: 8877: 8874: 8871: 8859: 8857: 8847: 8845: 8844:Euler numbers 8840: 8832: 8828: 8823: 8818: 8811: 8803: 8798: 8781: 8778: 8775: 8772: 8766: 8763: 8760: 8752: 8736: 8733: 8728: 8723: 8719: 8713: 8708: 8703: 8699: 8693: 8690: 8687: 8685: 8677: 8674: 8671: 8668: 8664: 8657: 8654: 8651: 8648: 8641: 8633: 8630: 8614: 8611: 8608: 8604: 8600: 8598: 8593: 8590: 8587: 8580: 8577: 8569: 8553: 8550: 8545: 8539: 8535: 8531: 8525: 8520: 8515: 8511: 8505: 8502: 8499: 8494: 8491: 8486: 8484: 8476: 8473: 8470: 8467: 8463: 8453: 8450: 8447: 8444: 8436: 8428: 8425: 8417: 8413: 8407: 8401: 8398: 8382: 8379: 8376: 8372: 8368: 8363: 8360: 8355: 8353: 8345: 8342: 8339: 8336: 8331: 8328: 8323: 8321: 8316: 8313: 8310: 8303: 8300: 8292: 8276: 8273: 8268: 8262: 8258: 8254: 8248: 8243: 8238: 8234: 8228: 8225: 8222: 8220: 8212: 8209: 8206: 8203: 8199: 8189: 8186: 8183: 8180: 8172: 8164: 8161: 8153: 8149: 8143: 8137: 8134: 8118: 8115: 8112: 8108: 8104: 8102: 8097: 8094: 8091: 8082: 8079: 8074: 8066: 8050: 8047: 8042: 8036: 8032: 8028: 8022: 8017: 8012: 8008: 8002: 7999: 7996: 7994: 7986: 7983: 7979: 7972: 7966: 7963: 7953: 7950: 7946: 7940: 7932: 7929: 7913: 7910: 7907: 7903: 7899: 7897: 7892: 7889: 7886: 7877: 7874: 7869: 7861: 7845: 7842: 7837: 7831: 7827: 7823: 7817: 7812: 7807: 7803: 7797: 7794: 7791: 7789: 7781: 7778: 7775: 7772: 7768: 7761: 7755: 7752: 7743: 7737: 7733: 7729: 7726: 7722: 7716: 7708: 7705: 7697: 7694: 7690: 7676: 7673: 7670: 7666: 7662: 7660: 7655: 7652: 7649: 7642: 7639:for all 7631: 7628: 7622: 7619: 7613: 7609: 7603: 7597: 7594: 7588: 7584: 7578: 7575: 7572: 7570: 7562: 7559: 7555: 7548: 7542: 7539: 7529: 7521: 7518: 7502: 7499: 7496: 7492: 7488: 7486: 7481: 7478: 7475: 7468: 7465:for all 7457: 7454: 7448: 7445: 7439: 7435: 7429: 7423: 7420: 7414: 7410: 7404: 7401: 7398: 7396: 7388: 7385: 7382: 7379: 7375: 7368: 7362: 7359: 7356: 7353: 7343: 7335: 7332: 7316: 7313: 7310: 7306: 7302: 7300: 7295: 7292: 7289: 7277: 7275: 7265: 7263: 7259: 7254: 7237: 7232: 7228: 7219: 7211: 7208: 7200: 7196: 7190: 7184: 7181: 7173: 7165: 7162: 7146: 7143: 7140: 7136: 7132: 7127: 7124: 7119: 7115: 7109: 7106: 7101: 7096: 7092: 7086: 7083: 7078: 7073: 7069: 7063: 7060: 7055: 7050: 7046: 7040: 7037: 7032: 7029: 7024: 7021: 7016: 7013: 7010: 7008: 6999: 6996: 6991: 6983: 6980: 6977: 6967: 6962: 6958: 6948: 6945: 6942: 6939: 6931: 6923: 6920: 6912: 6908: 6902: 6896: 6893: 6885: 6882: 6879: 6871: 6868: 6852: 6849: 6846: 6842: 6838: 6833: 6830: 6825: 6821: 6815: 6812: 6807: 6802: 6798: 6792: 6789: 6784: 6779: 6775: 6769: 6766: 6761: 6756: 6752: 6746: 6743: 6738: 6735: 6730: 6727: 6722: 6719: 6716: 6714: 6706: 6703: 6694: 6691: 6688: 6673: 6671: 6667: 6646: 6624: 6617: 6611: 6593: 6590: 6582: 6569: 6568:empty product 6563: 6557: 6544: 6538: 6535: 6527: 6524: 6521: 6518: 6515: 6509: 6503: 6500: 6497: 6491: 6485: 6480: 6476: 6473: 6470: 6467: 6464: 6456: 6451: 6448: 6445: 6441: 6437: 6426: 6423: 6407: 6406: 6401: 6386: 6382: 6370: 6367: 6349: 6346: 6343: 6339: 6335: 6330: 6322: 6319: 6316: 6305: 6303: 6293: 6291: 6275: 6272: 6264: 6250: 6233: 6228: 6225: 6222: 6218: 6212: 6208: 6202: 6199: 6196: 6180: 6177: 6174: 6170: 6166: 6164: 6154: 6146: 6143: 6140: 6133: 6122: 6119: 6116: 6112: 6108: 6098: 6095: 6092: 6088: 6084: 6082: 6072: 6064: 6061: 6058: 6051: 6040: 6036: 6025: 6022: 6019: 6015: 6011: 6009: 6001: 5998: 5995: 5991: 5977: 5975: 5965: 5961: 5954: 5946: 5939: 5921: 5918: 5910: 5897: 5893: 5889: 5884: 5867: 5864: 5861: 5856: 5851: 5847: 5841: 5836: 5831: 5827: 5821: 5818: 5815: 5810: 5805: 5801: 5793: 5790: 5787: 5779: 5776: 5763: 5760: 5757: 5753: 5749: 5747: 5739: 5736: 5733: 5727: 5724: 5717: 5714: 5711: 5706: 5701: 5697: 5691: 5686: 5681: 5677: 5671: 5668: 5665: 5662: 5657: 5652: 5648: 5635: 5632: 5629: 5625: 5621: 5618: 5616: 5608: 5605: 5602: 5596: 5593: 5581: 5578: 5573: 5563: 5548: 5544: 5537: 5534: 5528: 5524: 5511: 5508: 5505: 5501: 5497: 5491: 5488: 5484: 5480: 5477: 5471: 5468: 5460: 5458: 5454: 5449: 5431: 5428: 5425: 5419: 5416: 5410: 5406: 5400: 5394: 5391: 5385: 5381: 5375: 5372: 5369: 5366: 5363: 5357: 5354: 5348: 5344: 5331: 5328: 5325: 5321: 5317: 5312: 5308: 5299: 5296: 5277: 5273: 5265: 5255: 5249: 5245: 5240: 5231: 5223: 5213: 5211: 5206: 5193: 5187: 5179: 5176: 5173: 5169: 5165: 5161: 5157: 5151: 5148: 5145: 5139: 5126: 5122: 5114: 5110: 5103: 5095: 5089: 5086: 5083: 5077: 5069: 5061: 5052: 5048: 5039: 5033: 5028: 5025: 5015: 5011: 5007: 5002: 4997: 4984: 4978: 4975: 4968: 4960: 4956: 4952: 4940: 4937: 4934: 4931: 4925: 4915: 4912: 4909: 4905: 4899: 4895: 4891: 4888: 4884: 4876: 4872: 4865: 4857: 4851: 4848: 4845: 4839: 4831: 4828: 4815: 4810: 4807: 4804: 4800: 4793: 4790: 4784: 4780: 4767: 4764: 4761: 4757: 4751: 4748: 4744: 4740: 4735: 4731: 4725: 4714: 4711: 4705: 4701: 4688: 4685: 4682: 4678: 4669: 4665: 4661: 4655: 4645: 4644:Newton series 4629: 4618: 4605: 4599: 4596: 4593: 4587: 4584: 4577: 4573: 4565: 4559: 4554: 4549: 4535: 4532: 4526: 4522: 4509: 4506: 4503: 4499: 4491: 4487: 4480: 4467: 4463: 4458: 4454: 4446: 4443: 4433: 4430: 4426: 4422: 4417: 4409: 4404: 4402: 4398: 4397:Borel's lemma 4394: 4389: 4387: 4378: 4373: 4369: 4365: 4359: 4355: 4350: 4344: 4340: 4336: 4332: 4331:real analysis 4327: 4325: 4319: 4315: 4308: 4304: 4297: 4290: 4286: 4279: 4274: 4269: 4249: 4246: 4243: 4233: 4226: 4223: 4220: 4206: 4202: 4197: 4193: 4190: 4186: 4179: 4174: 4168: 4162: 4154: 4150: 4146: 4137: 4133: 4132:Fréchet space 4129: 4125: 4120: 4118: 4112: 4107: 4103: 4098: 4097: 4092: 4084: 4079: 4077: 4071: 4063: 4056: 4049: 4045: 4037: 4028: 4020: 4006: 4000: 3997: 3991: 3987: 3981: 3975: 3972: 3966: 3962: 3956: 3950: 3947: 3941: 3937: 3931: 3928: 3925: 3921: 3917: 3914: 3906: 3902: 3896: 3884: 3880:(black). For 3877: 3870: 3861: 3854: 3846: 3839: 3831: 3826: 3813: 3810: 3806: 3802: 3799: 3795: 3791: 3788: 3784: 3783:complex plane 3780: 3776: 3772: 3769: 3766: 3762: 3761: 3760: 3757: 3755: 3739: 3727: 3723: 3719: 3715: 3711: 3707: 3702: 3698: 3694: 3684: 3680: 3674: 3667: 3652: 3648: 3644: 3638: 3635: 3627: 3616: 3609: 3597: 3594: 3590: 3576: 3563: 3558: 3550: 3547: 3544: 3536: 3532: 3521: 3518: 3515: 3511: 3507: 3501: 3495: 3487: 3477: 3467: 3463: 3452: 3445: 3440:The function 3438: 3433: 3423: 3421: 3416: 3414: 3409: 3407: 3403: 3398: 3384: 3378: 3375: 3369: 3364: 3360: 3356: 3353: 3350: 3330: 3318: 3314: 3308: 3296: 3293: 3285: 3281: 3254: 3251: 3248: 3242: 3239: 3222: 3219: 3213: 3210: 3206: 3185: 3164: 3161: 3158: 3154: 3135: 3132: 3129: 3126: 3107: 3104: 3101: 3098: 3079: 3076: 3073: 3070: 3062: 3044: 3041: 3038: 3035: 3014: 3011: 3008: 3005: 2986: 2983: 2980: 2977: 2957: 2954: 2951: 2948: 2938: 2934: 2933:James Gregory 2929: 2927: 2923: 2919: 2915: 2911: 2907: 2903: 2898: 2896: 2892: 2888: 2884: 2880: 2876: 2872: 2869: 2859: 2855: 2848: 2841: 2835: 2825: 2819: 2802: 2799: 2796: 2791: 2786: 2782: 2776: 2771: 2766: 2762: 2756: 2751: 2746: 2742: 2736: 2731: 2726: 2722: 2716: 2713: 2710: 2707: 2704: 2702: 2694: 2691: 2685: 2682: 2676: 2672: 2666: 2660: 2657: 2651: 2647: 2641: 2635: 2632: 2626: 2622: 2616: 2610: 2607: 2601: 2597: 2591: 2585: 2582: 2576: 2572: 2566: 2560: 2557: 2551: 2547: 2541: 2539: 2531: 2528: 2522: 2518: 2505: 2502: 2499: 2495: 2482: 2479: 2475: 2470: 2457: 2454: 2451: 2446: 2441: 2436: 2432: 2429: 2426: 2422: 2411: 2407: 2403: 2398: 2392: 2389: 2386: 2378: 2375: 2370: 2367: 2364: 2361: 2353: 2346: 2339: 2326: 2323: 2320: 2315: 2307: 2304: 2301: 2292: 2289: 2283: 2278: 2270: 2267: 2264: 2255: 2252: 2246: 2241: 2233: 2230: 2227: 2218: 2215: 2209: 2203: 2200: 2197: 2186: 2182: 2176: 2169: 2156: 2153: 2150: 2145: 2141: 2134: 2131: 2125: 2120: 2116: 2109: 2106: 2100: 2095: 2091: 2084: 2081: 2075: 2072: 2069: 2061: 2059: 2049: 2042: 2029: 2026: 2023: 2018: 2010: 2007: 2004: 1998: 1993: 1985: 1982: 1979: 1973: 1967: 1964: 1961: 1955: 1952: 1944: 1940: 1930: 1914: 1903: 1890: 1887: 1884: 1879: 1875: 1871: 1866: 1862: 1858: 1855: 1852: 1849: 1841: 1840: 1831: 1814: 1812: 1802: 1789: 1784: 1780: 1773: 1770: 1762: 1751: 1744: 1730: 1727: 1724: 1720: 1716: 1713: 1710: 1705: 1701: 1694: 1691: 1683: 1676: 1673: 1666: 1661: 1657: 1650: 1647: 1639: 1632: 1629: 1622: 1619: 1613: 1610: 1602: 1595: 1592: 1585: 1579: 1573: 1563: 1558: 1554: 1545: 1541: 1519: 1509: 1505: 1496: 1490: 1473: 1468: 1460: 1457: 1454: 1445: 1442: 1434: 1423: 1416: 1402: 1399: 1396: 1392: 1388: 1385: 1382: 1377: 1369: 1366: 1363: 1354: 1351: 1343: 1336: 1333: 1326: 1321: 1313: 1310: 1307: 1298: 1295: 1287: 1280: 1277: 1270: 1264: 1261: 1258: 1249: 1246: 1238: 1231: 1228: 1221: 1215: 1209: 1202: 1197: 1193: 1189: 1185: 1179: 1175: 1171: 1167: 1157: 1152:) containing 1151: 1150:complex plane 1147: 1143: 1142:open interval 1135: 1131: 1127: 1123: 1119: 1111: 1099: 1093: 1088: 1083: 1081: 1077: 1073: 1069: 1065: 1061: 1057: 1053: 1052:Taylor series 1049: 1037: 1032: 1030: 1025: 1023: 1018: 1017: 1015: 1014: 1007: 1004: 1002: 999: 997: 994: 992: 989: 987: 984: 982: 979: 977: 974: 973: 965: 964: 957: 954: 952: 949: 947: 944: 942: 939: 938: 930: 929: 918: 915: 913: 910: 908: 905: 904: 903: 902: 892: 891: 880: 877: 875: 872: 870: 867: 865: 862: 860: 859:Line integral 857: 855: 852: 850: 847: 846: 845: 844: 840: 839: 834: 831: 829: 826: 824: 821: 819: 816: 815: 814: 813: 809: 808: 802: 801:Multivariable 796: 795: 784: 781: 779: 776: 774: 771: 769: 766: 764: 761: 759: 756: 755: 754: 753: 749: 748: 743: 740: 738: 735: 733: 730: 728: 725: 723: 720: 718: 715: 714: 713: 712: 706: 700: 699: 688: 685: 683: 680: 678: 675: 673: 670: 668: 664: 662: 659: 657: 654: 652: 649: 647: 644: 642: 639: 638: 637: 636: 633: 630: 629: 624: 621: 619: 616: 614: 611: 609: 606: 604: 601: 598: 594: 591: 590: 589: 588: 582: 576: 575: 564: 561: 559: 556: 554: 551: 549: 546: 543: 539: 536: 534: 531: 528: 524: 520: 519:trigonometric 516: 513: 511: 508: 506: 503: 501: 498: 497: 496: 495: 491: 490: 485: 482: 480: 477: 475: 472: 470: 467: 464: 460: 457: 455: 452: 451: 450: 449: 445: 444: 439: 436: 434: 431: 429: 426: 425: 424: 423: 417: 411: 410: 399: 396: 394: 391: 389: 386: 384: 381: 379: 376: 374: 371: 369: 366: 364: 361: 359: 356: 354: 351: 350: 349: 348: 345: 342: 341: 336: 333: 331: 330:Related rates 328: 326: 323: 321: 318: 316: 313: 311: 308: 307: 306: 305: 301: 300: 293: 290: 288: 287:of a function 285: 283: 282:infinitesimal 280: 279: 278: 275: 272: 268: 265: 264: 263: 262: 258: 257: 251: 245: 244: 238: 235: 233: 230: 228: 225: 224: 219: 216: 214: 211: 210: 206: 203: 202: 201: 200: 181: 175: 172: 166: 160: 157: 154: 151: 144: 137: 134: 128: 123: 119: 110: 109: 106: 103: 102: 98: 97: 89: 83: 77: 71: 65: 59: 53: 47: 41: 34: 30: 19: 19937:Power series 19931: 19679:Lucas number 19631:Powers of 10 19611:Cubic number 19522: 19500: 19480: 19461: 19452: 19447: 19426: 19406: 19394: 19383:. Retrieved 19376:the original 19363: 19359: 19315: 19311: 19292: 19265: 19240: 19211: 19182:(1): 33–45. 19179: 19175: 19166: 19159:Hille, Einar 19138: 19109: 19081: 19077: 19052: 19048: 19039: 19014: 19009:Merzbach, U. 18989: 18969: 18958:. New York: 18953: 18912: 18877: 18857: 18841: 18825: 18818:Hofmann 1939 18813: 18764: 18752: 18728: 18716: 18685: 18657: 18645: 18629: 18613: 18601: 18596:, See §8.9.. 18589: 18573: 18473: 18469: 18445: 18439: 18433: 18429: 18422: 18418: 18387: 18258: 18250: 18241: 18237: 17957: 17954: 17591: 17588: 17328: 17325: 17017: 17014: 16936: 16932: 16928: 16920: 16916: 16912: 16908: 16904: 16885: 16743: 16728: 16724: 16721: 16715: 16711: 16696: 16692: 16689: 16686: 16481: 16478: 16460: 16139: 16132: 16128: 16081: 15040: 15037: 15027: 15024:power series 15021: 15006: 14983: 14343: 14340: 14170: 14164: 14154: 14150: 14146: 14014: 13900: 13627: 13557: 13108: 13079: 12932: 12929: 12837: 12834: 12710: 12707: 12645: 12642: 12630: 12275: 12271: 12146: 12143: 12032: 11943: 11861: 11858: 11745: 11742: 11725: 11428: 11425: 11136: 11130:The regular 11129: 10934: 10927: 10623: 10616: 10604: 10334: 10327: 10325: 10095: 10088: 9906: 9899: 9884: 9874: 9872:The numbers 9871: 8860: 8853: 8838: 8830: 8826: 8816: 8806: 8799: 7278: 7271: 7255: 6674: 6644: 6622: 6615: 6612: 6561: 6558: 6408: 6402: 6306: 6299: 6251: 5978: 5971: 5959: 5952: 5944: 5937: 5895: 5885: 5582: 5569: 5461: 5457:Bell numbers 5450: 5300: 5261: 5253: 5247: 5225: 5207: 5070: 5059: 5050: 5046: 5037: 5031: 5013: 5009: 5005: 4998: 4832: 4829: 4670: 4663: 4659: 4656: 4627: 4619: 4468: 4461: 4439: 4428: 4424: 4420: 4405: 4393:coefficients 4390: 4376: 4357: 4353: 4348: 4342: 4338: 4328: 4317: 4313: 4306: 4302: 4295: 4288: 4284: 4277: 4270: 4155: 4148: 4144: 4121: 4110: 4105: 4101: 4094: 4090: 4082: 4080: 4075: 4069: 4061: 4054: 4050: 4043: 4035: 4026: 4021: 3907: 3900: 3894: 3890: 3882: 3875: 3859: 3852: 3844: 3758: 3732:is far from 3700: 3682: 3678: 3675: 3668: 3598: 3592: 3588: 3577: 3488: 3465: 3461: 3458: 3450: 3443: 3417: 3413:Brook Taylor 3410: 3405: 3401: 3399: 3277:integral of 3061:Isaac Newton 2937:John Collins 2930: 2899: 2871:Zeno of Elea 2865: 2853: 2846: 2839: 2833: 2823: 2820: 2483: 2477: 2471: 2354: 2344: 2340: 2187: 2180: 2174: 2170: 2062: 2056:denotes the 2047: 2043: 1945: 1938: 1928: 1912: 1904: 1842: 1829: 1815: 1808: 1561: 1543: 1539: 1512:denotes the 1507: 1503: 1493:denotes the 1488: 1201:power series 1195: 1177: 1173: 1163: 1109: 1091: 1084: 1075: 1072:Brook Taylor 1064:infinite sum 1055: 1051: 1045: 622: 515:Substitution 277:Differential 250:Differential 87: 81: 75: 69: 63: 57: 51: 45: 39: 29: 19804:Conditional 19792:Convergence 19783:Telescoping 19768:Alternating 19684:Pell number 19007:Boyer, C.; 18757:Feller 2003 18736:Feller 2003 18697:Struik 1969 18693:Taylor 1715 18578:Banner 2007 18478:could be a 16888:multi-index 7258:linear term 6666:square root 5574:(with base 5292:(with base 4457:Einar Hille 4408:singularity 4366:studied in 3710:square root 3059:derived by 1536:itself and 1136:at a point 1116:increases. 1087:partial sum 1068:derivatives 1048:mathematics 976:Precalculus 969:Miscellanea 934:Specialized 841:Definitions 608:Alternating 446:Definitions 259:Definitions 20048:Categories 19829:Convergent 19773:Convergent 19385:2023-02-18 18938:References 18903:, p. 18884:, p. 18864:, p. 18848:, p. 18832:, p. 18802:, p. 18793:, p. 18771:, p. 18721:Rudin 1980 18701:Bruce 2007 18676:Malet 1993 18636:, p. 18620:, p. 18618:Kline 1990 18580:, p. 18256:, we have 16939:) = (0, 0) 16473:See also: 7272:The usual 5220:See also: 4386:continuous 4128:meager set 4124:convergent 3789:available. 3704:, and the 2922:arctangent 2887:Democritus 2883:Archimedes 1811:polynomial 1518:derivative 1182:, that is 1160:Definition 1122:convergent 1100:of degree 1098:polynomial 956:Variations 951:Stochastic 941:Fractional 810:Formalisms 773:Divergence 742:Identities 722:Divergence 267:Derivative 218:Continuity 19860:Divergent 19778:Divergent 19640:Advanced 19616:Factorial 19564:Sequences 19524:MathWorld 19507:EMS Press 19340:120101519 19206:(2002) . 19128:818811840 19108:(2003) . 19098:122105736 19069:120553186 19049:Resonance 18650:Dani 2012 18492:uniformly 18488:pointwise 18340:⋯ 18312:− 18279:⁡ 18220:⋯ 18192:− 18167:⋯ 18141:− 18126:− 18111:− 18096:− 18068:− 18030:− 18009:− 17955:produces 17937:⋯ 17885:− 17839:− 17824:− 17768:− 17696:− 17653:− 17495:− 17202:− 17160:⁡ 17059:⁡ 16983:⁡ 16847:α 16842:∂ 16825:α 16818:α 16805:− 16786:≥ 16778:α 16769:∑ 16670:⋯ 16656:− 16594:− 16531:− 16399:− 16353:− 16338:− 16282:− 16217:− 16174:− 16064:⋯ 16045:− 16016:− 15987:− 15958:∂ 15945:∂ 15932:∂ 15911:… 15883:∂ 15859:∑ 15838:∑ 15817:∑ 15773:− 15744:− 15715:∂ 15702:∂ 15681:… 15653:∂ 15629:∑ 15608:∑ 15573:− 15544:∂ 15523:… 15501:∂ 15478:∑ 15455:… 15407:… 15359:∂ 15356:⋯ 15331:∂ 15308:⋯ 15291:∂ 15263:⋯ 15218:− 15202:⋯ 15172:− 15151:∞ 15129:∑ 15125:⋯ 15120:∞ 15098:∑ 15071:… 15009:operators 14923:∞ 14908:∑ 14859:∞ 14844:∑ 14797:− 14757:∞ 14742:∑ 14717:− 14693:∞ 14678:∑ 14647:∞ 14632:∑ 14582:∞ 14567:∑ 14536:∞ 14521:∑ 14478:∞ 14463:∑ 14432:∞ 14417:∑ 14324:⋯ 14207:∞ 14192:∑ 14125:⋯ 14038:⁡ 13996:⁡ 13884:… 13813:− 13754:− 13695:− 13581:⁡ 13540:⋯ 13477:− 13424:− 13371:− 13309:⋯ 13306:− 13256:− 13240:⋯ 13091:⁡ 13066:⋯ 12956:⁡ 12913:⋯ 12910:− 12860:− 12848:⁡ 12818:⋯ 12684:⁡ 12566:− 12546:− 12526:− 12494:− 12488:⁡ 12454:− 12448:⁡ 12411:− 12405:⁡ 12384:− 12375:− 12369:⁡ 12337:− 12331:⁡ 12309:⁡ 12212:− 12172:− 12163:− 12157:⁡ 12067:− 12043:⁡ 12011:− 12005:⁡ 11999:↦ 11961:⁡ 11955:↦ 11914:− 11908:⁡ 11829:π 11813:π 11807:− 11796:∈ 11780:⁡ 11771:⁡ 11699:− 11685:− 11671:∞ 11656:∏ 11625:∞ 11610:∑ 11547:ϑ 11543:− 11518:ϑ 11491:− 11471:ϑ 11437:ϑ 11399:− 11385:∞ 11370:∏ 11339:∞ 11324:∑ 11304:− 11258:ϑ 11254:− 11229:ϑ 11202:− 11182:ϑ 11165:− 11145:ϑ 11082:− 11074:∞ 11059:∑ 11024:ϑ 10997:∞ 10982:∑ 10947:ϑ 10856:− 10812:∞ 10797:∑ 10772:π 10680:∞ 10665:∑ 10640:π 10525:− 10514:∞ 10499:∑ 10401:− 10390:∞ 10375:∑ 10253:∞ 10238:∑ 10212:χ 10149:∞ 10134:∑ 10108:χ 10042:∞ 10027:∑ 9962:∞ 9947:∑ 9851:± 9848:≠ 9833:≤ 9816:for 9809:⋯ 9713:∞ 9698:∑ 9684:⁡ 9671:≤ 9654:for 9647:⋯ 9644:− 9599:− 9486:− 9475:∞ 9460:∑ 9446:⁡ 9431:π 9409:for 9402:⋯ 9374:− 9329:− 9310:− 9268:− 9222:∞ 9207:∑ 9193:⁡ 9172:⋯ 9069:∞ 9054:∑ 9040:⁡ 9019:⋯ 8904:∞ 8889:∑ 8875:⁡ 8779:± 8776:≠ 8761:≤ 8744:for 8737:⋯ 8734:− 8694:− 8631:− 8620:∞ 8605:∑ 8591:⁡ 8578:≤ 8561:for 8554:⋯ 8551:− 8526:− 8506:− 8500:− 8492:π 8388:∞ 8373:∑ 8369:− 8361:π 8343:⁡ 8337:− 8329:π 8314:⁡ 8301:≤ 8284:for 8277:⋯ 8124:∞ 8109:∑ 8095:⁡ 8080:π 8058:for 8051:⋯ 7930:− 7919:∞ 7904:∑ 7890:⁡ 7875:π 7853:for 7846:⋯ 7779:− 7730:− 7706:− 7682:∞ 7667:∑ 7653:⁡ 7632:⋯ 7629:− 7579:− 7519:− 7508:∞ 7493:∑ 7479:⁡ 7458:⋯ 7455:− 7405:− 7333:− 7322:∞ 7307:∑ 7293:⁡ 7163:− 7152:∞ 7137:∑ 7128:⋯ 7102:− 7056:− 7017:− 6992:− 6946:− 6883:− 6869:− 6858:∞ 6843:∑ 6834:⋯ 6831:− 6785:− 6739:− 6664:give the 6519:− 6516:α 6510:⋯ 6501:− 6498:α 6492:α 6468:− 6465:α 6442:∏ 6424:α 6368:α 6355:∞ 6340:∑ 6331:α 6226:− 6200:− 6186:∞ 6171:∑ 6144:− 6120:− 6104:∞ 6089:∑ 6062:− 6031:∞ 6016:∑ 5999:− 5865:⋯ 5862:− 5822:− 5777:− 5769:∞ 5754:∑ 5728:⁡ 5715:⋯ 5712:− 5692:− 5672:− 5666:− 5641:∞ 5626:∑ 5622:− 5606:− 5597:⁡ 5517:∞ 5502:∑ 5489:− 5481:⁡ 5472:⁡ 5429:⋯ 5337:∞ 5322:∑ 5135:∞ 5130:∞ 5127:− 5123:∫ 5107:→ 5068:. Hence, 4921:∞ 4906:∑ 4889:− 4869:→ 4773:∞ 4758:∑ 4749:− 4722:Δ 4694:∞ 4679:∑ 4546:Δ 4515:∞ 4500:∑ 4484:→ 4351:equal to 4224:≠ 4191:− 4091:remainder 3982:− 3932:− 3926:≈ 3918:⁡ 3714:logarithm 3548:− 3527:∞ 3512:∑ 3385:π 3370:− 3357:⁡ 3297:⁡ 3249:π 3214:⁡ 3162:⁡ 3130:⁡ 3102:⁡ 3074:⁡ 3042:⁡ 3009:⁡ 2981:⁡ 2952:⁡ 2879:Aristotle 2877:. Later, 2800:⋯ 2695:⋯ 2511:∞ 2496:∑ 2455:⋯ 2430:− 2399:− 2390:− 2365:⁡ 2324:⋯ 2305:− 2284:− 2268:− 2231:− 2210:− 2201:− 2154:⋯ 2151:− 2126:− 2101:− 2076:− 2070:− 2027:⋯ 2008:− 1999:− 1983:− 1965:− 1956:− 1888:⋯ 1736:∞ 1721:∑ 1714:⋯ 1495:factorial 1458:− 1408:∞ 1393:∑ 1386:⋯ 1367:− 1311:− 1262:− 1199:, is the 1146:open disk 946:Malliavin 833:Geometric 732:Laplacian 682:Dirichlet 593:Geometric 173:− 120:∫ 20033:Category 19799:Absolute 19405:(1980). 19332:41133959 19165:(1957). 19011:(1991). 18952:(1970). 18671:Roy 1990 18511:See also 18472: ( 18458:interval 18421: ( 16907: ( 16727: ( 16703:gradient 16695: ( 9888:are the 8820:are the 5018:, where 5008: ( 4423: ( 4356: ( 4341: ( 4316: ( 4305: ( 4287: ( 4240:if 4217:if 4147: ( 4096:residual 4068:−1 < 4042:−1 < 4034:−π < 3851:−1 < 3738:diverges 3726:converge 3681: ( 3476:analytic 3464: ( 2831:is also 2052:, where 2046:ln(1 − 1805:Examples 1677:‴ 1633:″ 1596:′ 1337:‴ 1281:″ 1232:′ 1176: ( 1134:analytic 1060:function 986:Glossary 896:Advanced 874:Jacobian 828:Exterior 758:Gradient 750:Theorems 717:Gradient 656:Integral 618:Binomial 603:Harmonic 463:improper 459:Integral 416:Integral 398:Reynolds 373:Quotient 302:Concepts 138:′ 105:Calculus 19809:Uniform 19509:, 2001 19196:3028095 18638:202–203 18406:cosines 18240:ln(1 + 16919:ln(1 + 16894:Example 16733:is the 16701:is the 8802:radians 6670:inverse 6661:⁠ 6649:⁠ 6639:⁠ 6627:⁠ 5951:ln(1 + 5936:ln(1 − 5455:of the 5065:⁠ 5042:⁠ 4634:is the 4442:bounded 4130:in the 4053:ln(1 + 3874:ln(1 + 3843:ln(1 + 3596:gives: 2895:Liu Hui 2862:History 1933:⁠ 1919:⁠ 1837:is the 1834:⁠ 1819:⁠ 1148:in the 1128:of the 981:History 879:Hessian 768:Stokes' 763:Green's 595: ( 517: ( 461: ( 383:Inverse 358:Product 269: ( 19761:Series 19568:series 19468: 19434: 19413: 19338: 19330: 19299: 19280: 19249: 19226: 19194: 19147: 19126: 19116: 19096: 19067: 19023: 18996: 18977: 18410:powers 18248:| 18238:Since 16687:where 14341:Thus, 13881: 13800: 13797: 13741: 13738: 13682: 13679: 13657: 13654: 9842: 9681:artanh 9443:arsinh 8824:. The 8770: 8588:arctan 8340:arcsin 8311:arccos 8092:arcsin 4464:> 0 4414:; see 4293:about 4046:< 1 4038:< π 4024:| 3885:> 1 3862:> 1 3857:. For 3722:arctan 3716:, the 3712:, the 3693:entire 3354:arctan 3294:arcsec 3071:arctan 3006:arcsin 2920:, and 2918:cosine 2837:, and 1486:Here, 1062:is an 1050:, the 823:Tensor 818:Matrix 705:Vector 623:Taylor 581:Series 213:Limits 79:, and 19704:array 19584:Basic 19451:[ 19379:(PDF) 19356:(PDF) 19336:S2CID 19328:JSTOR 19192:JSTOR 19094:S2CID 19065:S2CID 18566:Notes 18402:sines 14149:(1 + 9883:tanh 6613:When 5022:is a 4620:Here 4449:(0,∞) 4083:error 4076:worse 3275:(the 2924:(see 1186:at a 1126:limit 1058:of a 646:Ratio 613:Power 527:Euler 505:Discs 500:Parts 368:Power 363:Chain 292:total 19644:list 19566:and 19466:ISBN 19432:ISBN 19411:ISBN 19297:ISBN 19278:ISBN 19247:ISBN 19224:ISBN 19145:ISBN 19124:OCLC 19114:ISBN 19021:ISBN 18994:ISBN 18975:ISBN 18624:–37. 18404:and 18360:< 16915:) = 16719:and 16121:and 13984:and 11988:and 10928:The 10089:The 9900:The 9426:< 9190:tanh 9037:cosh 8872:sinh 8854:The 8842:are 8837:sec 8815:tan 8075:< 7870:< 6642:and 6618:= −1 6591:< 6559:(If 6300:The 6273:< 5972:The 5947:= −1 5919:< 5570:The 5262:The 5242:The 5208:The 4427:) = 4081:The 3893:sin 3748:and 3343:and 3027:and 2914:sine 2866:The 2849:− 0) 2352:is: 1911:1 − 1909:for 1828:1 − 1548:and 1188:real 1166:real 1144:(or 1085:The 727:Curl 687:Abel 651:Root 38:sin 19368:doi 19320:doi 19270:doi 19216:doi 19184:doi 19086:doi 19057:doi 18795:252 18582:530 18443:at 16705:of 16137:is 15015:or 14035:cos 13993:cos 13578:cos 13088:cos 12953:cos 12845:cos 12681:cos 12485:cos 12445:cos 12402:cos 12366:cos 12328:cos 12227:720 12154:cos 12002:cos 11905:cos 11777:cos 10605:In 9394:315 7887:sec 7650:tan 7476:cos 7290:sin 7110:256 7087:128 6816:256 6793:128 6647:= − 6564:= 0 5964:.) 5962:= 1 5478:exp 5469:exp 5256:+ 1 5100:lim 5003:of 4862:lim 4477:lim 4447:on 4349:not 4329:In 4298:= 0 4280:= 0 4275:at 4271:is 4134:of 4093:or 4072:≤ 1 4064:= 0 3915:sin 3903:= 0 3855:≤ 1 3773:An 3740:at 3728:if 3676:If 3459:If 3453:= 0 3286:), 3279:sec 3211:tan 3186:tan 3159:sec 3127:sec 3099:tan 3039:cot 2978:cos 2949:sin 2912:of 2792:120 2481:is 2343:ln 2185:is 2183:= 1 2178:at 2173:ln 1943:is 1941:= 1 1936:at 1564:= 0 1520:of 1516:th 1497:of 1190:or 1168:or 1108:th 1094:+ 1 1054:or 1046:In 353:Sum 90:= 0 85:at 20050:: 19521:. 19505:, 19499:, 19364:63 19362:. 19358:. 19334:. 19326:. 19316:46 19314:. 19276:. 19264:. 19222:. 19210:. 19190:. 19180:14 19178:. 19161:; 19122:. 19092:. 19082:34 19080:. 19063:. 19053:17 19051:. 19038:. 18948:; 18905:85 18893:^ 18886:75 18870:81 18868:, 18866:75 18850:15 18834:14 18804:15 18780:^ 18773:69 18622:35 18482:.) 18432:= 18363:1. 18276:ln 17571:1. 17157:ln 17056:ln 16980:ln 16935:, 16714:= 16465:. 16131:, 15028:as 15019:. 14168:: 13872:24 13847:24 12581:45 12561:12 12306:ln 12207:24 12040:ln 12030:) 12014:1. 11958:ln 11885:ln 11768:ln 11601:24 11574:16 11551:01 11522:00 11475:01 11441:00 11315:24 11285:16 11262:01 11233:00 11186:01 11149:00 11028:01 10951:00 10869:16 10719:16 10472:Ti 10348:Ti 10332:: 10000:Li 9920:Li 9892:. 9639:40 9380:17 9369:15 8846:. 8546:40 8269:40 8043:24 7838:15 7264:. 7107:63 7084:35 7064:16 6770:16 6672:: 6625:= 6610:. 5725:ln 5594:ln 5448:. 5230:. 5032:jh 5012:+ 4466:, 4326:. 4119:. 3800:). 3591:= 3408:. 3207:ln 3198:), 3155:ln 2916:, 2772:24 2362:ln 2060:: 2054:ln 1550:0! 1542:− 525:, 521:, 82:13 76:11 73:, 67:, 61:, 55:, 49:, 19646:) 19642:( 19556:e 19549:t 19542:v 19527:. 19474:. 19440:. 19419:. 19388:. 19370:: 19351:π 19342:. 19322:: 19305:. 19286:. 19272:: 19255:. 19232:. 19218:: 19198:. 19186:: 19153:. 19130:. 19100:. 19088:: 19071:. 19059:: 19042:. 19029:. 19002:. 18983:. 18907:. 18888:. 18872:. 18852:. 18836:. 18820:. 18775:. 18652:. 18640:. 18584:. 18476:) 18474:x 18470:f 18446:a 18440:f 18434:a 18430:x 18425:) 18423:x 18419:f 18400:( 18356:| 18352:y 18348:| 18343:, 18337:+ 18332:2 18328:y 18321:2 18318:1 18309:y 18306:x 18303:+ 18300:y 18297:= 18294:) 18291:y 18288:+ 18285:1 18282:( 18271:x 18267:e 18251:y 18244:) 18242:y 18217:+ 18212:2 18208:y 18201:2 18198:1 18189:y 18186:x 18183:+ 18180:y 18177:= 18164:+ 18159:) 18152:2 18148:) 18144:0 18138:y 18135:( 18132:) 18129:1 18123:( 18120:+ 18117:) 18114:0 18108:y 18105:( 18102:) 18099:0 18093:x 18090:( 18087:2 18084:+ 18079:2 18075:) 18071:0 18065:x 18062:( 18059:0 18054:( 18047:2 18044:1 18039:+ 18036:) 18033:0 18027:y 18024:( 18021:1 18018:+ 18015:) 18012:0 18006:x 18003:( 18000:0 17997:+ 17994:0 17991:= 17984:) 17981:y 17978:, 17975:x 17972:( 17969:T 17934:+ 17930:) 17926:) 17923:b 17920:, 17917:a 17914:( 17909:y 17906:y 17902:f 17896:2 17892:) 17888:b 17882:y 17879:( 17876:+ 17873:) 17870:b 17867:, 17864:a 17861:( 17856:y 17853:x 17849:f 17845:) 17842:b 17836:y 17833:( 17830:) 17827:a 17821:x 17818:( 17815:2 17812:+ 17809:) 17806:b 17803:, 17800:a 17797:( 17792:x 17789:x 17785:f 17779:2 17775:) 17771:a 17765:x 17762:( 17758:( 17751:! 17748:2 17744:1 17739:+ 17727:) 17724:b 17721:, 17718:a 17715:( 17710:y 17706:f 17702:) 17699:b 17693:y 17690:( 17687:+ 17684:) 17681:b 17678:, 17675:a 17672:( 17667:x 17663:f 17659:) 17656:a 17650:x 17647:( 17644:+ 17641:) 17638:b 17635:, 17632:a 17629:( 17626:f 17621:= 17618:) 17615:y 17612:, 17609:x 17606:( 17603:T 17568:= 17565:) 17562:0 17559:, 17556:0 17553:( 17548:x 17545:y 17541:f 17537:= 17530:) 17527:0 17524:, 17521:0 17518:( 17513:y 17510:x 17506:f 17498:1 17492:= 17485:) 17482:0 17479:, 17476:0 17473:( 17468:y 17465:y 17461:f 17453:0 17450:= 17443:) 17440:0 17437:, 17434:0 17431:( 17426:x 17423:x 17419:f 17411:1 17408:= 17401:) 17398:0 17395:, 17392:0 17389:( 17384:y 17380:f 17372:0 17369:= 17362:) 17359:0 17356:, 17353:0 17350:( 17345:x 17341:f 17308:. 17302:y 17299:+ 17296:1 17290:x 17286:e 17280:= 17275:x 17272:y 17268:f 17264:= 17255:y 17252:x 17248:f 17235:2 17231:) 17227:y 17224:+ 17221:1 17218:( 17212:x 17208:e 17199:= 17190:y 17187:y 17183:f 17175:) 17172:y 17169:+ 17166:1 17163:( 17152:x 17148:e 17144:= 17135:x 17132:x 17128:f 17117:y 17114:+ 17111:1 17105:x 17101:e 17095:= 17086:y 17082:f 17074:) 17071:y 17068:+ 17065:1 17062:( 17051:x 17047:e 17043:= 17034:x 17030:f 17001:, 16998:) 16995:y 16992:+ 16989:1 16986:( 16975:x 16971:e 16967:= 16964:) 16961:y 16958:, 16955:x 16952:( 16949:f 16937:b 16933:a 16931:( 16923:) 16921:y 16917:e 16913:y 16911:, 16909:x 16905:f 16872:, 16869:) 16865:a 16861:( 16857:) 16853:f 16835:( 16828:! 16814:) 16809:a 16801:x 16797:( 16789:0 16782:| 16774:| 16765:= 16762:) 16758:x 16754:( 16751:T 16731:) 16729:a 16725:f 16722:D 16716:a 16712:x 16707:f 16699:) 16697:a 16693:f 16690:D 16673:, 16667:+ 16664:) 16660:a 16652:x 16648:( 16644:} 16640:) 16636:a 16632:( 16629:f 16624:2 16620:D 16615:{ 16608:T 16603:) 16598:a 16590:x 16586:( 16580:! 16577:2 16573:1 16568:+ 16565:) 16561:a 16557:( 16554:f 16551:D 16545:T 16540:) 16535:a 16527:x 16523:( 16520:+ 16517:) 16513:a 16509:( 16506:f 16503:= 16500:) 16496:x 16492:( 16489:T 16445:) 16440:) 16437:b 16434:, 16431:a 16428:( 16423:y 16420:y 16416:f 16410:2 16406:) 16402:b 16396:y 16393:( 16390:+ 16387:) 16384:b 16381:, 16378:a 16375:( 16370:y 16367:x 16363:f 16359:) 16356:b 16350:y 16347:( 16344:) 16341:a 16335:x 16332:( 16329:2 16326:+ 16323:) 16320:b 16317:, 16314:a 16311:( 16306:x 16303:x 16299:f 16293:2 16289:) 16285:a 16279:x 16276:( 16271:( 16263:! 16260:2 16256:1 16251:+ 16248:) 16245:b 16242:, 16239:a 16236:( 16231:y 16227:f 16223:) 16220:b 16214:y 16211:( 16208:+ 16205:) 16202:b 16199:, 16196:a 16193:( 16188:x 16184:f 16180:) 16177:a 16171:x 16168:( 16165:+ 16162:) 16159:b 16156:, 16153:a 16150:( 16147:f 16135:) 16133:b 16129:a 16127:( 16123:y 16119:x 16105:) 16102:y 16099:, 16096:x 16093:( 16090:f 16061:+ 16058:) 16053:l 16049:a 16040:l 16036:x 16032:( 16029:) 16024:k 16020:a 16011:k 16007:x 16003:( 16000:) 15995:j 15991:a 15982:j 15978:x 15974:( 15966:l 15962:x 15953:k 15949:x 15940:j 15936:x 15927:) 15922:d 15918:a 15914:, 15908:, 15903:1 15899:a 15895:( 15892:f 15887:3 15874:d 15869:1 15866:= 15863:l 15853:d 15848:1 15845:= 15842:k 15832:d 15827:1 15824:= 15821:j 15810:! 15807:3 15803:1 15798:+ 15786:) 15781:k 15777:a 15768:k 15764:x 15760:( 15757:) 15752:j 15748:a 15739:j 15735:x 15731:( 15723:k 15719:x 15710:j 15706:x 15697:) 15692:d 15688:a 15684:, 15678:, 15673:1 15669:a 15665:( 15662:f 15657:2 15644:d 15639:1 15636:= 15633:k 15623:d 15618:1 15615:= 15612:j 15601:! 15598:2 15594:1 15589:+ 15586:) 15581:j 15577:a 15568:j 15564:x 15560:( 15552:j 15548:x 15539:) 15534:d 15530:a 15526:, 15520:, 15515:1 15511:a 15507:( 15504:f 15493:d 15488:1 15485:= 15482:j 15474:+ 15471:) 15466:d 15462:a 15458:, 15452:, 15447:1 15443:a 15439:( 15436:f 15433:= 15423:) 15418:d 15414:a 15410:, 15404:, 15399:1 15395:a 15391:( 15387:) 15377:d 15373:n 15367:d 15363:x 15349:1 15345:n 15339:1 15335:x 15326:f 15319:d 15315:n 15311:+ 15305:+ 15300:1 15296:n 15284:( 15276:! 15271:d 15267:n 15260:! 15255:1 15251:n 15241:d 15237:n 15232:) 15226:d 15222:a 15213:d 15209:x 15205:( 15195:1 15191:n 15186:) 15180:1 15176:a 15167:1 15163:x 15159:( 15146:0 15143:= 15138:d 15134:n 15115:0 15112:= 15107:1 15103:n 15094:= 15087:) 15082:d 15078:x 15074:, 15068:, 15063:1 15059:x 15055:( 15052:T 14961:. 14956:n 14952:x 14945:! 14942:n 14937:1 14934:+ 14931:n 14918:0 14915:= 14912:n 14904:= 14892:n 14888:x 14881:! 14878:n 14873:1 14870:+ 14867:n 14854:1 14851:= 14848:n 14840:+ 14837:1 14834:= 14822:n 14818:x 14813:) 14806:! 14803:) 14800:1 14794:n 14791:( 14787:1 14782:+ 14776:! 14773:n 14769:1 14763:( 14752:1 14749:= 14746:n 14738:+ 14735:1 14732:= 14726:! 14723:) 14720:1 14714:n 14711:( 14705:n 14701:x 14688:1 14685:= 14682:n 14674:+ 14668:! 14665:n 14659:n 14655:x 14642:1 14639:= 14636:n 14628:+ 14625:1 14622:= 14609:! 14606:n 14600:1 14597:+ 14594:n 14590:x 14577:0 14574:= 14571:n 14563:+ 14557:! 14554:n 14548:n 14544:x 14531:1 14528:= 14525:n 14517:+ 14514:1 14511:= 14505:! 14502:n 14496:1 14493:+ 14490:n 14486:x 14473:0 14470:= 14467:n 14459:+ 14453:! 14450:n 14444:n 14440:x 14427:0 14424:= 14421:n 14413:= 14408:x 14404:e 14400:x 14397:+ 14392:x 14388:e 14384:= 14375:x 14371:e 14367:) 14364:x 14361:+ 14358:1 14355:( 14327:. 14321:+ 14315:! 14312:4 14306:4 14302:x 14296:+ 14290:! 14287:3 14281:3 14277:x 14271:+ 14265:! 14262:2 14256:2 14252:x 14246:+ 14243:x 14240:+ 14237:1 14234:= 14228:! 14225:n 14219:n 14215:x 14202:0 14199:= 14196:n 14188:= 14183:x 14179:e 14165:e 14160:x 14155:e 14153:) 14151:x 14128:. 14122:+ 14117:4 14113:x 14106:2 14103:1 14097:+ 14092:3 14088:x 14081:3 14078:2 14072:+ 14067:2 14063:x 14059:+ 14056:x 14053:+ 14050:1 14047:= 14041:x 14029:x 14025:e 14011:: 13999:x 13970:x 13966:e 13945:) 13942:x 13939:( 13936:g 13914:i 13910:c 13887:. 13878:, 13869:1 13863:= 13858:0 13854:c 13844:1 13838:+ 13833:2 13829:c 13822:2 13819:1 13808:4 13804:c 13794:, 13788:6 13785:1 13779:= 13774:1 13770:c 13763:2 13760:1 13749:3 13745:c 13735:, 13729:2 13726:1 13720:= 13715:0 13711:c 13704:2 13701:1 13690:2 13686:c 13676:, 13673:1 13670:= 13665:1 13661:c 13651:, 13648:1 13645:= 13640:0 13636:c 13614:, 13609:x 13605:e 13584:x 13575:) 13572:x 13569:( 13566:g 13537:+ 13532:4 13528:x 13523:) 13516:! 13513:4 13507:0 13503:c 13497:+ 13492:2 13487:2 13483:c 13472:4 13468:c 13463:( 13459:+ 13454:3 13450:x 13445:) 13439:2 13434:1 13430:c 13419:3 13415:c 13410:( 13406:+ 13401:2 13397:x 13392:) 13386:2 13381:0 13377:c 13366:2 13362:c 13357:( 13353:+ 13350:x 13345:1 13341:c 13337:+ 13332:0 13328:c 13324:= 13313:) 13300:! 13297:4 13291:4 13287:x 13281:+ 13275:! 13272:2 13266:2 13262:x 13253:1 13249:( 13244:) 13237:+ 13232:4 13228:x 13222:4 13218:c 13214:+ 13209:3 13205:x 13199:3 13195:c 13191:+ 13186:2 13182:x 13176:2 13172:c 13168:+ 13165:x 13160:1 13156:c 13152:+ 13147:0 13143:c 13138:( 13134:= 13125:x 13121:e 13094:x 13063:+ 13058:4 13054:x 13048:4 13044:c 13040:+ 13035:3 13031:x 13025:3 13021:c 13017:+ 13012:2 13008:x 13002:2 12998:c 12994:+ 12991:x 12986:1 12982:c 12978:+ 12973:0 12969:c 12965:= 12959:x 12947:x 12943:e 12916:. 12904:! 12901:4 12895:4 12891:x 12885:+ 12879:! 12876:2 12870:2 12866:x 12857:1 12854:= 12851:x 12821:, 12815:+ 12809:! 12806:4 12800:4 12796:x 12790:+ 12784:! 12781:3 12775:3 12771:x 12765:+ 12759:! 12756:2 12750:2 12746:x 12740:+ 12737:x 12734:+ 12731:1 12728:= 12723:x 12719:e 12693:. 12687:x 12675:x 12671:e 12665:= 12662:) 12659:x 12656:( 12653:g 12612:. 12607:) 12602:8 12598:x 12594:( 12589:O 12586:+ 12576:6 12572:x 12556:4 12552:x 12541:2 12536:2 12532:x 12523:= 12511:) 12505:4 12501:) 12497:1 12491:x 12482:( 12478:( 12473:O 12470:+ 12465:3 12461:) 12457:1 12451:x 12442:( 12436:3 12433:1 12427:+ 12422:2 12418:) 12414:1 12408:x 12399:( 12393:2 12390:1 12381:) 12378:1 12372:x 12363:( 12360:= 12348:) 12343:) 12340:1 12334:x 12325:( 12322:+ 12319:1 12314:( 12303:= 12296:) 12293:x 12290:( 12287:f 12258:. 12253:) 12248:8 12244:x 12240:( 12235:O 12232:+ 12222:6 12218:x 12202:4 12198:x 12192:+ 12187:2 12182:2 12178:x 12169:= 12166:1 12160:x 12128:) 12123:4 12119:x 12115:( 12110:O 12107:+ 12102:3 12097:3 12093:x 12087:+ 12082:2 12077:2 12073:x 12064:x 12061:= 12058:) 12055:x 12052:+ 12049:1 12046:( 12008:x 11996:x 11976:) 11973:x 11970:+ 11967:1 11964:( 11952:x 11930:, 11925:) 11920:) 11917:1 11911:x 11902:( 11899:+ 11896:1 11891:( 11881:= 11878:) 11875:x 11872:( 11869:f 11845:, 11840:) 11832:2 11823:, 11816:2 11801:( 11793:x 11789:, 11786:) 11783:x 11774:( 11765:= 11762:) 11759:x 11756:( 11753:f 11702:1 11696:k 11693:2 11689:x 11682:1 11678:1 11666:1 11663:= 11660:k 11652:= 11647:n 11643:x 11639:) 11636:n 11633:( 11630:Q 11620:0 11617:= 11614:n 11606:= 11597:/ 11593:1 11587:] 11578:x 11567:4 11563:) 11559:x 11556:( 11538:4 11534:) 11530:x 11527:( 11509:[ 11502:3 11498:/ 11494:1 11487:) 11483:x 11480:( 11465:6 11461:/ 11457:1 11453:) 11449:x 11446:( 11407:k 11403:x 11396:1 11392:1 11380:1 11377:= 11374:k 11366:= 11361:n 11357:x 11353:) 11350:n 11347:( 11344:P 11334:0 11331:= 11328:n 11320:= 11311:/ 11307:1 11298:] 11289:x 11278:4 11274:) 11270:x 11267:( 11249:4 11245:) 11241:x 11238:( 11220:[ 11213:3 11209:/ 11205:2 11198:) 11194:x 11191:( 11176:6 11172:/ 11168:1 11161:) 11157:x 11154:( 11108:2 11104:n 11099:x 11093:n 11089:) 11085:1 11079:( 11069:1 11066:= 11063:n 11055:2 11052:+ 11049:1 11046:= 11039:) 11036:x 11033:( 11012:2 11008:n 11003:x 10992:1 10989:= 10986:n 10978:2 10975:+ 10972:1 10969:= 10962:) 10959:x 10956:( 10908:n 10905:2 10901:x 10892:4 10888:) 10884:! 10881:n 10878:( 10873:n 10865:) 10862:n 10859:2 10853:1 10850:( 10843:2 10839:] 10835:! 10832:) 10829:n 10826:2 10823:( 10820:[ 10807:0 10804:= 10801:n 10793:= 10786:) 10783:x 10780:( 10777:E 10769:2 10758:n 10755:2 10751:x 10742:4 10738:) 10734:! 10731:n 10728:( 10723:n 10711:2 10707:] 10703:! 10700:) 10697:n 10694:2 10691:( 10688:[ 10675:0 10672:= 10669:n 10661:= 10654:) 10651:x 10648:( 10645:K 10637:2 10585:1 10582:+ 10579:n 10576:2 10572:x 10563:3 10559:) 10555:1 10552:+ 10549:n 10546:2 10543:( 10536:n 10532:) 10528:1 10522:( 10509:0 10506:= 10503:n 10495:= 10488:) 10485:x 10482:( 10477:3 10461:1 10458:+ 10455:n 10452:2 10448:x 10439:2 10435:) 10431:1 10428:+ 10425:n 10422:2 10419:( 10412:n 10408:) 10404:1 10398:( 10385:0 10382:= 10379:n 10371:= 10364:) 10361:x 10358:( 10353:2 10306:1 10303:+ 10300:n 10297:2 10293:x 10284:3 10280:) 10276:1 10273:+ 10270:n 10267:2 10264:( 10260:1 10248:0 10245:= 10242:n 10234:= 10227:) 10224:x 10221:( 10216:3 10202:1 10199:+ 10196:n 10193:2 10189:x 10180:2 10176:) 10172:1 10169:+ 10166:n 10163:2 10160:( 10156:1 10144:0 10141:= 10138:n 10130:= 10123:) 10120:x 10117:( 10112:2 10069:n 10065:x 10057:3 10053:n 10049:1 10037:1 10034:= 10031:n 10023:= 10016:) 10013:x 10010:( 10005:3 9989:n 9985:x 9977:2 9973:n 9969:1 9957:1 9954:= 9951:n 9943:= 9936:) 9933:x 9930:( 9925:2 9885:x 9877:k 9875:B 9854:1 9845:x 9839:, 9836:1 9829:| 9825:x 9821:| 9806:+ 9801:5 9796:5 9792:x 9786:+ 9781:3 9776:3 9772:x 9766:+ 9763:x 9760:= 9749:1 9746:+ 9743:n 9740:2 9734:1 9731:+ 9728:n 9725:2 9721:x 9708:0 9705:= 9702:n 9694:= 9687:x 9674:1 9667:| 9663:x 9659:| 9633:5 9629:x 9625:3 9619:+ 9614:6 9609:3 9605:x 9596:x 9593:= 9583:1 9580:+ 9577:n 9574:2 9570:x 9563:) 9560:1 9557:+ 9554:n 9551:2 9548:( 9543:2 9539:) 9535:! 9532:n 9529:( 9524:n 9520:4 9514:! 9511:) 9508:n 9505:2 9502:( 9497:n 9493:) 9489:1 9483:( 9470:0 9467:= 9464:n 9456:= 9449:x 9434:2 9422:| 9418:x 9414:| 9399:+ 9388:7 9384:x 9363:5 9359:x 9355:2 9349:+ 9344:3 9339:3 9335:x 9326:x 9323:= 9313:1 9307:n 9304:2 9300:x 9293:! 9290:) 9287:n 9284:2 9281:( 9275:) 9271:1 9263:n 9259:4 9254:( 9248:n 9244:4 9238:n 9235:2 9231:B 9217:1 9214:= 9211:n 9203:= 9196:x 9183:x 9169:+ 9163:! 9160:4 9154:4 9150:x 9144:+ 9138:! 9135:2 9129:2 9125:x 9119:+ 9116:1 9113:= 9102:! 9099:) 9096:n 9093:2 9090:( 9084:n 9081:2 9077:x 9064:0 9061:= 9058:n 9050:= 9043:x 9030:x 9016:+ 9010:! 9007:5 9001:5 8997:x 8991:+ 8985:! 8982:3 8976:3 8972:x 8966:+ 8963:x 8960:= 8949:! 8946:) 8943:1 8940:+ 8937:n 8934:2 8931:( 8925:1 8922:+ 8919:n 8916:2 8912:x 8899:0 8896:= 8893:n 8885:= 8878:x 8839:x 8831:k 8827:E 8817:x 8809:k 8807:B 8782:i 8773:x 8767:, 8764:1 8757:| 8753:x 8749:| 8729:5 8724:5 8720:x 8714:+ 8709:3 8704:3 8700:x 8691:x 8688:= 8678:1 8675:+ 8672:n 8669:2 8665:x 8658:1 8655:+ 8652:n 8649:2 8642:n 8638:) 8634:1 8628:( 8615:0 8612:= 8609:n 8601:= 8594:x 8581:1 8574:| 8570:x 8566:| 8540:5 8536:x 8532:3 8521:6 8516:3 8512:x 8503:x 8495:2 8487:= 8477:1 8474:+ 8471:n 8468:2 8464:x 8457:) 8454:1 8451:+ 8448:n 8445:2 8442:( 8437:2 8433:) 8429:! 8426:n 8423:( 8418:n 8414:4 8408:! 8405:) 8402:n 8399:2 8396:( 8383:0 8380:= 8377:n 8364:2 8356:= 8346:x 8332:2 8324:= 8317:x 8304:1 8297:| 8293:x 8289:| 8274:+ 8263:5 8259:x 8255:3 8249:+ 8244:6 8239:3 8235:x 8229:+ 8226:x 8223:= 8213:1 8210:+ 8207:n 8204:2 8200:x 8193:) 8190:1 8187:+ 8184:n 8181:2 8178:( 8173:2 8169:) 8165:! 8162:n 8159:( 8154:n 8150:4 8144:! 8141:) 8138:n 8135:2 8132:( 8119:0 8116:= 8113:n 8105:= 8098:x 8083:2 8071:| 8067:x 8063:| 8048:+ 8037:4 8033:x 8029:5 8023:+ 8018:2 8013:2 8009:x 8003:+ 8000:1 7997:= 7987:n 7984:2 7980:x 7973:! 7970:) 7967:n 7964:2 7961:( 7954:n 7951:2 7947:E 7941:n 7937:) 7933:1 7927:( 7914:0 7911:= 7908:n 7900:= 7893:x 7878:2 7866:| 7862:x 7858:| 7843:+ 7832:5 7828:x 7824:2 7818:+ 7813:3 7808:3 7804:x 7798:+ 7795:x 7792:= 7782:1 7776:n 7773:2 7769:x 7762:! 7759:) 7756:n 7753:2 7750:( 7744:) 7738:n 7734:4 7727:1 7723:( 7717:n 7713:) 7709:4 7703:( 7698:n 7695:2 7691:B 7677:1 7674:= 7671:n 7663:= 7656:x 7643:x 7623:! 7620:4 7614:4 7610:x 7604:+ 7598:! 7595:2 7589:2 7585:x 7576:1 7573:= 7563:n 7560:2 7556:x 7549:! 7546:) 7543:n 7540:2 7537:( 7530:n 7526:) 7522:1 7516:( 7503:0 7500:= 7497:n 7489:= 7482:x 7469:x 7449:! 7446:5 7440:5 7436:x 7430:+ 7424:! 7421:3 7415:3 7411:x 7402:x 7399:= 7389:1 7386:+ 7383:n 7380:2 7376:x 7369:! 7366:) 7363:1 7360:+ 7357:n 7354:2 7351:( 7344:n 7340:) 7336:1 7330:( 7317:0 7314:= 7311:n 7303:= 7296:x 7238:. 7233:n 7229:x 7220:2 7216:) 7212:! 7209:n 7206:( 7201:n 7197:4 7191:! 7188:) 7185:n 7182:2 7179:( 7174:n 7170:) 7166:1 7160:( 7147:0 7144:= 7141:n 7133:= 7125:+ 7120:5 7116:x 7097:4 7093:x 7079:+ 7074:3 7070:x 7061:5 7051:2 7047:x 7041:8 7038:3 7033:+ 7030:x 7025:2 7022:1 7014:1 7011:= 7000:2 6997:1 6988:) 6984:x 6981:+ 6978:1 6975:( 6968:, 6963:n 6959:x 6952:) 6949:1 6943:n 6940:2 6937:( 6932:2 6928:) 6924:! 6921:n 6918:( 6913:n 6909:4 6903:! 6900:) 6897:n 6894:2 6891:( 6886:1 6880:n 6876:) 6872:1 6866:( 6853:0 6850:= 6847:n 6839:= 6826:5 6822:x 6813:7 6808:+ 6803:4 6799:x 6790:5 6780:3 6776:x 6767:1 6762:+ 6757:2 6753:x 6747:8 6744:1 6736:x 6731:2 6728:1 6723:+ 6720:1 6717:= 6707:2 6704:1 6699:) 6695:x 6692:+ 6689:1 6686:( 6658:2 6655:/ 6652:1 6645:α 6636:2 6633:/ 6630:1 6623:α 6616:α 6608:α 6594:1 6587:| 6583:x 6579:| 6562:n 6545:. 6539:! 6536:n 6531:) 6528:1 6525:+ 6522:n 6513:( 6507:) 6504:1 6495:( 6486:= 6481:k 6477:1 6474:+ 6471:k 6457:n 6452:1 6449:= 6446:k 6438:= 6432:) 6427:n 6419:( 6387:n 6383:x 6376:) 6371:n 6363:( 6350:0 6347:= 6344:n 6336:= 6327:) 6323:x 6320:+ 6317:1 6314:( 6276:1 6269:| 6265:x 6261:| 6234:. 6229:2 6223:n 6219:x 6213:2 6209:n 6206:) 6203:1 6197:n 6194:( 6181:2 6178:= 6175:n 6167:= 6155:3 6151:) 6147:x 6141:1 6138:( 6134:1 6123:1 6117:n 6113:x 6109:n 6099:1 6096:= 6093:n 6085:= 6073:2 6069:) 6065:x 6059:1 6056:( 6052:1 6041:n 6037:x 6026:0 6023:= 6020:n 6012:= 6002:x 5996:1 5992:1 5960:x 5955:) 5953:x 5945:x 5940:) 5938:x 5922:1 5915:| 5911:x 5907:| 5868:. 5857:3 5852:3 5848:x 5842:+ 5837:2 5832:2 5828:x 5819:x 5816:= 5811:n 5806:n 5802:x 5794:1 5791:+ 5788:n 5784:) 5780:1 5774:( 5764:1 5761:= 5758:n 5750:= 5743:) 5740:x 5737:+ 5734:1 5731:( 5718:, 5707:3 5702:3 5698:x 5687:2 5682:2 5678:x 5669:x 5663:= 5658:n 5653:n 5649:x 5636:1 5633:= 5630:n 5619:= 5612:) 5609:x 5603:1 5600:( 5577:e 5549:n 5545:x 5538:! 5535:n 5529:n 5525:B 5512:0 5509:= 5506:n 5498:= 5495:) 5492:1 5485:x 5475:( 5446:x 5432:. 5426:+ 5420:! 5417:3 5411:3 5407:x 5401:+ 5395:! 5392:2 5386:2 5382:x 5376:+ 5373:x 5370:+ 5367:1 5364:= 5358:! 5355:n 5349:n 5345:x 5332:0 5329:= 5326:n 5318:= 5313:x 5309:e 5295:e 5278:x 5274:e 5254:n 5248:e 5228:x 5194:. 5191:) 5188:x 5185:( 5180:h 5177:, 5174:h 5170:/ 5166:t 5162:P 5158:d 5155:) 5152:x 5149:+ 5146:a 5143:( 5140:f 5115:+ 5111:0 5104:h 5096:= 5093:) 5090:t 5087:+ 5084:a 5081:( 5078:f 5062:! 5060:j 5056:/ 5053:) 5051:h 5049:/ 5047:t 5045:( 5040:· 5038:e 5020:X 5016:) 5014:X 5010:a 5006:f 4985:. 4979:! 4976:j 4969:j 4965:) 4961:h 4957:/ 4953:t 4950:( 4944:) 4941:h 4938:j 4935:+ 4932:a 4929:( 4926:f 4916:0 4913:= 4910:j 4900:h 4896:/ 4892:t 4885:e 4877:+ 4873:0 4866:h 4858:= 4855:) 4852:t 4849:+ 4846:a 4843:( 4840:f 4816:. 4811:j 4808:+ 4805:i 4801:a 4794:! 4791:j 4785:j 4781:u 4768:0 4765:= 4762:j 4752:u 4745:e 4741:= 4736:i 4732:a 4726:n 4715:! 4712:n 4706:n 4702:u 4689:0 4686:= 4683:n 4664:i 4660:a 4652:a 4648:f 4640:h 4636:n 4628:h 4622:Δ 4606:. 4603:) 4600:t 4597:+ 4594:a 4591:( 4588:f 4585:= 4578:n 4574:h 4569:) 4566:a 4563:( 4560:f 4555:n 4550:h 4536:! 4533:n 4527:n 4523:t 4510:0 4507:= 4504:n 4492:+ 4488:0 4481:h 4462:t 4429:e 4425:x 4421:f 4412:x 4382:z 4377:e 4360:) 4358:x 4354:f 4345:) 4343:x 4339:f 4320:) 4318:x 4314:f 4309:) 4307:x 4303:f 4296:x 4291:) 4289:x 4285:f 4278:x 4250:0 4247:= 4244:x 4234:0 4227:0 4221:x 4207:2 4203:x 4198:/ 4194:1 4187:e 4180:{ 4175:= 4172:) 4169:x 4166:( 4163:f 4151:) 4149:x 4145:f 4140:f 4113:) 4111:x 4109:( 4106:n 4102:R 4087:n 4070:x 4062:a 4057:) 4055:x 4044:x 4036:x 4027:x 4007:. 4001:! 3998:7 3992:7 3988:x 3976:! 3973:5 3967:5 3963:x 3957:+ 3951:! 3948:3 3942:3 3938:x 3929:x 3922:x 3901:x 3895:x 3883:x 3878:) 3876:x 3860:x 3853:x 3847:) 3845:x 3811:. 3750:b 3746:x 3742:x 3734:b 3730:x 3701:e 3689:x 3685:) 3683:x 3679:f 3671:b 3653:n 3649:a 3645:= 3639:! 3636:n 3631:) 3628:b 3625:( 3620:) 3617:n 3614:( 3610:f 3593:b 3589:x 3584:n 3580:x 3564:. 3559:n 3555:) 3551:b 3545:x 3542:( 3537:n 3533:a 3522:0 3519:= 3516:n 3508:= 3505:) 3502:x 3499:( 3496:f 3484:f 3480:x 3472:b 3468:) 3466:x 3462:f 3451:x 3444:e 3379:2 3376:1 3365:x 3361:e 3351:2 3331:, 3326:) 3319:x 3315:e 3309:2 3302:( 3260:) 3255:x 3252:+ 3243:2 3240:1 3232:( 3223:2 3220:1 3165:x 3136:, 3133:x 3108:, 3105:x 3080:, 3077:x 3057:) 3045:x 3036:x 3015:, 3012:x 2987:, 2984:x 2958:, 2955:x 2941:( 2856:! 2854:n 2847:x 2845:( 2840:e 2834:e 2829:x 2824:e 2803:. 2797:+ 2787:5 2783:x 2777:+ 2767:4 2763:x 2757:+ 2752:6 2747:3 2743:x 2737:+ 2732:2 2727:2 2723:x 2717:+ 2714:x 2711:+ 2708:1 2705:= 2692:+ 2686:! 2683:5 2677:5 2673:x 2667:+ 2661:! 2658:4 2652:4 2648:x 2642:+ 2636:! 2633:3 2627:3 2623:x 2617:+ 2611:! 2608:2 2602:2 2598:x 2592:+ 2586:! 2583:1 2577:1 2573:x 2567:+ 2561:! 2558:0 2552:0 2548:x 2542:= 2532:! 2529:n 2523:n 2519:x 2506:0 2503:= 2500:n 2478:e 2458:. 2452:+ 2447:2 2442:2 2437:) 2433:a 2427:x 2423:( 2412:2 2408:a 2404:1 2396:) 2393:a 2387:x 2384:( 2379:a 2376:1 2371:+ 2368:a 2350:a 2345:x 2327:, 2321:+ 2316:4 2312:) 2308:1 2302:x 2299:( 2293:4 2290:1 2279:3 2275:) 2271:1 2265:x 2262:( 2256:3 2253:1 2247:+ 2242:2 2238:) 2234:1 2228:x 2225:( 2219:2 2216:1 2207:) 2204:1 2198:x 2195:( 2181:a 2175:x 2157:. 2146:4 2142:x 2135:4 2132:1 2121:3 2117:x 2110:3 2107:1 2096:2 2092:x 2085:2 2082:1 2073:x 2050:) 2048:x 2030:. 2024:+ 2019:3 2015:) 2011:1 2005:x 2002:( 1994:2 1990:) 1986:1 1980:x 1977:( 1974:+ 1971:) 1968:1 1962:x 1959:( 1953:1 1939:a 1929:x 1925:/ 1922:1 1913:x 1907:x 1891:. 1885:+ 1880:3 1876:x 1872:+ 1867:2 1863:x 1859:+ 1856:x 1853:+ 1850:1 1830:x 1825:/ 1822:1 1790:. 1785:n 1781:x 1774:! 1771:n 1766:) 1763:0 1760:( 1755:) 1752:n 1749:( 1745:f 1731:0 1728:= 1725:n 1717:= 1711:+ 1706:3 1702:x 1695:! 1692:3 1687:) 1684:0 1681:( 1674:f 1667:+ 1662:2 1658:x 1651:! 1648:2 1643:) 1640:0 1637:( 1630:f 1623:+ 1620:x 1614:! 1611:1 1606:) 1603:0 1600:( 1593:f 1586:+ 1583:) 1580:0 1577:( 1574:f 1562:a 1546:) 1544:a 1540:x 1538:( 1534:f 1530:f 1526:a 1522:f 1514:n 1510:) 1508:a 1506:( 1504:f 1499:n 1491:! 1489:n 1474:. 1469:n 1465:) 1461:a 1455:x 1452:( 1446:! 1443:n 1438:) 1435:a 1432:( 1427:) 1424:n 1421:( 1417:f 1403:0 1400:= 1397:n 1389:= 1383:+ 1378:3 1374:) 1370:a 1364:x 1361:( 1355:! 1352:3 1347:) 1344:a 1341:( 1334:f 1327:+ 1322:2 1318:) 1314:a 1308:x 1305:( 1299:! 1296:2 1291:) 1288:a 1285:( 1278:f 1271:+ 1268:) 1265:a 1259:x 1256:( 1250:! 1247:1 1242:) 1239:a 1236:( 1229:f 1222:+ 1219:) 1216:a 1213:( 1210:f 1196:a 1180:) 1178:x 1174:f 1154:x 1138:x 1114:n 1106:n 1102:n 1092:n 1035:e 1028:t 1021:v 599:) 544:) 540:( 529:) 465:) 273:) 185:) 182:a 179:( 176:f 170:) 167:b 164:( 161:f 158:= 155:t 152:d 148:) 145:t 142:( 135:f 129:b 124:a 92:. 88:x 70:9 64:7 58:5 52:3 46:1 40:x 20:)

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