Editing Taylor series (section)

==List of Maclaurin series of some common functions==
{{see also|List of mathematical series}}
Several important Maclaurin series expansions follow. All these expansions are valid for complex arguments {{mvar|x}}.

=== Exponential function ===
[[File:Exp series.gif|right|thumb|The [[exponential function]] {{math|''e''<sup>''x''</sup>}} (in blue), and the sum of the first {{math|''n'' + 1}} terms of its Taylor series at 0 (in red).]]
The [[exponential function]] <math>e^x</math> (with base {{mvar|[[e (mathematics)|e]]}}) has Maclaurin series{{sfn|Abramowitz|Stegun|1970|p=[https://books.google.com/books?id=MtU8uP7XMvoC&pg=PA69 69]}}

<math display="block"> e^{x} = \sum^{\infty}_{n=0} \frac{x^n}{n!} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots. </math>
It converges for all {{mvar|x}}.

The exponential [[generating function]] of the [[Bell number]]s is the exponential function of the predecessor of the exponential function:

<math display="block">\exp(\exp{x}-1) = \sum_{n=0}^{\infty} \frac{B_n}{n!}x^{n}</math>

=== Natural logarithm ===
The [[natural logarithm]] (with base {{mvar|[[e (mathematics)|e]]}}) has Maclaurin series<ref name="bileodeau-abramowitz">{{multiref
 |{{harvnb|Bilodeau|Thie|Keough|2010|p=[https://books.google.com/books?id=nsHisqNlsuIC&pg=PA252 252]}}
 |{{harvnb|Abramowitz|Stegun|1970|p=[https://books.google.com/books?id=MtU8uP7XMvoC&pg=PA15 15]}}
}}</ref>

<math display="block"> \begin{align}
\ln(1-x) &= - \sum^{\infty}_{n=1} \frac{x^n}n = -x - \frac{x^2}2 - \frac{x^3}3 - \cdots , \\
\ln(1+x) &= \sum^\infty_{n=1} (-1)^{n+1}\frac{x^n}n = x - \frac{x^2}2 + \frac{x^3}3 - \cdots .
\end{align}</math>

The last series is known as [[Mercator series]], named after [[Nicholas Mercator]] (since it was published in his 1668 treatise ''Logarithmotechnia'').{{sfn|Hofmann|1939}} Both of these series converge for <math>|x| < 1</math>. (In addition, the series for {{math|ln(1 − ''x'')}} converges for {{math|''x'' {{=}} −1}}, and the series for {{math|ln(1 + ''x'')}} converges for {{math|''x'' {{=}} 1}}.)<ref name="bileodeau-abramowitz" />

=== Geometric series ===

The [[geometric series]] and its derivatives have Maclaurin series

<math display="block">\begin{align}
\frac{1}{1-x} &= \sum^\infty_{n=0} x^n \\
\frac{1}{(1-x)^2} &= \sum^\infty_{n=1} nx^{n-1} \\
\frac{1}{(1-x)^3} &= \sum^\infty_{n=2} \frac{(n-1)n}{2} x^{n-2}.
\end{align}</math>

All are convergent for <math>|x| < 1</math>. These are special cases of the [[#Binomial series|binomial series]] given in the next section.

=== Binomial series ===
The [[binomial series]] is the power series

<math display="block">(1+x)^\alpha = \sum_{n=0}^\infty \binom{\alpha}{n} x^n</math>

whose coefficients are the generalized [[binomial coefficient]]s{{sfn|Abramowitz|Stegun|1970|p=[https://books.google.com/books?id=MtU8uP7XMvoC&pg=PA14 14]}}

<math display="block">\binom{\alpha}{n} = \prod_{k=1}^n \frac{\alpha-k+1}k = \frac{\alpha(\alpha-1)\cdots(\alpha-n+1)}{n!}.</math>

(If {{math|1= ''n'' = 0}}, this product is an [[empty product]] and has value 1.)  It converges for <math>|x| < 1</math> for any real or complex number {{mvar|α}}.

When {{math|1=''α'' = −1}}, this is essentially the infinite geometric series mentioned in the previous section.  The special cases {{math|1=''α'' = {{sfrac|1|2}}}} and {{math|1=''α'' = −{{sfrac|1|2}}}} give the [[square root]] function and its [[multiplicative inverse|inverse]]:{{sfn|Abramowitz|Stegun|1970|p=[https://books.google.com/books?id=MtU8uP7XMvoC&pg=PA15 15]}}

<math display="block">\begin{align} 
(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^{\infty}_{n=0} \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^{\infty}_{n=0} \frac{(-1)^n(2n)!}{4^n (n!)^2} x^n.
\end{align}
</math>

When only the [[linear approximation|linear term]] is retained, this simplifies to the [[binomial approximation]].

=== Trigonometric functions ===
The usual [[trigonometric function]]s and their inverses have the following Maclaurin series:{{sfn|Abramowitz|Stegun|1970|p=[https://books.google.com/books?id=MtU8uP7XMvoC&pg=PA75 75], [https://books.google.com/books?id=MtU8uP7XMvoC&pg=PA81 81]}}

<math display="block">\begin{align}
\sin x &= \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n+1)!} x^{2n+1} &&= x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots && \text{for all } x\\[6pt]
\cos x &= \sum^{\infty}_{n=0} \frac{(-1)^n}{(2n)!} x^{2n} &&= 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots && \text{for all } x\\[6pt]
\tan x &= \sum^{\infty}_{n=1} \frac{B_{2n} (-4)^n \left(1-4^n\right)}{(2n)!} x^{2n-1} &&= x + \frac{x^3}{3} + \frac{2 x^5}{15} + \cdots && \text{for }|x| < \frac{\pi}{2}\\[6pt]
\sec x &= \sum^{\infty}_{n=0} \frac{(-1)^n E_{2n}}{(2n)!} x^{2n} &&=1+\frac{x^2}{2}+\frac{5x^4}{24}+\cdots && \text{for }|x| < \frac{\pi}{2}\\[6pt]
\arcsin x &= \sum^{\infty}_{n=0} \frac{(2n)!}{4^n (n!)^2 (2n+1)} x^{2n+1} &&=x+\frac{x^3}{6}+\frac{3x^5}{40}+\cdots && \text{for }|x| \le 1\\[6pt]
\arccos x &=\frac{\pi}{2}-\arcsin x\\&=\frac{\pi}{2}- \sum^{\infty}_{n=0} \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| \le 1\\[6pt]
\arctan x &= \sum^{\infty}_{n=0} \frac{(-1)^n}{2n+1} x^{2n+1} &&=x-\frac{x^3}{3} + \frac{x^5}{5}-\cdots && \text{for }|x| \le 1,\ x\neq\pm i
\end{align}</math>

All angles are expressed in [[radian]]s. The numbers {{math|''B<sub>k</sub>''}} appearing in the expansions of {{math|tan ''x''}} are the [[Bernoulli numbers]]. The {{math|''E''<sub>''k''</sub>}} in the expansion of {{math|sec ''x''}} are [[Euler number]]s.{{sfn|Abramowitz|Stegun|1970|p=[https://books.google.com/books?id=MtU8uP7XMvoC&pg=PA75 75]}}

=== Hyperbolic functions ===
The [[hyperbolic function]]s have Maclaurin series closely related to the series for the corresponding trigonometric functions:{{sfn|Abramowitz|Stegun|1970|p=[https://books.google.com/books?id=MtU8uP7XMvoC&pg=PA85 85]}}

<math display="block">\begin{align}
\sinh x &= \sum^{\infty}_{n=0} \frac{x^{2n+1}}{(2n+1)!} &&= x + \frac{x^3}{3!} + \frac{x^5}{5!} + \cdots && \text{for all } x\\[6pt]
\cosh x &= \sum^{\infty}_{n=0} \frac{x^{2n}}{(2n)!} &&= 1 + \frac{x^2}{2!} + \frac{x^4}{4!} + \cdots && \text{for all } x\\[6pt]
\tanh x &= \sum^{\infty}_{n=1} \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}\\[6pt]
\operatorname{arsinh} x &= \sum^{\infty}_{n=0} \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| \le 1\\[6pt]
\operatorname{artanh} x &= \sum^{\infty}_{n=0} \frac{x^{2n+1}}{2n+1} &&=x + \frac{x^3}{3} + \frac{x^5}{5}  +\cdots && \text{for }|x| \le 1,\ x\neq\pm 1
\end{align}</math>

The numbers {{math|''B<sub>k</sub>''}} appearing in the series for {{math|tanh ''x''}} are the [[Bernoulli numbers]].{{sfn|Abramowitz|Stegun|1970|p=[https://books.google.com/books?id=MtU8uP7XMvoC&pg=PA85 85]}}

=== Polylogarithmic functions ===
The [[polylogarithm]]s have these defining identities:

<math display="block">\begin{align}
\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{align}</math>

The [[Legendre chi function]]s are defined as follows:

<math display="block">\begin{align}
\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{align}</math>

And the formulas presented below are called ''[[inverse tangent integral]]s'':

<math display="block">\begin{align}
\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{align}</math>

In [[Statistical mechanics|statistical thermodynamics]] these formulas are of great importance.

=== Elliptic functions ===

The complete [[elliptic integral]]s of first kind K and of second kind E can be defined as follows:

<math display="block">\begin{align}
\frac{2}{\pi}K(x) &= \sum_{n = 0}^{\infty} \frac{[(2n)!]^2}{16^{n}(n!)^4}x^{2n} \\
\frac{2}{\pi}E(x) &= \sum_{n = 0}^{\infty} \frac{[(2n)!]^2}{(1 - 2n)16^{n}(n!)^4}x^{2n}
\end{align}</math>

The [[Theta function|Jacobi theta functions]] describe the world of the elliptic modular functions and they have these Taylor series:

<math display="block">\begin{align}
\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{align}</math>

The regular [[Partition function (number theory)|partition number sequence]] P(n) has this generating function:

<math display="block">\vartheta_{00}(x)^{-1/6}\vartheta_{01}(x)^{-2/3}\biggl[\frac{\vartheta_{00}(x)^4 - \vartheta_{01}(x)^4}{16\,x}\biggr]^{-1/24} = \sum_{n=0}^{\infty} P(n)x^n = \prod_{k = 1}^{\infty} \frac{1}{1 - x^{k}}</math> 

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

<math display="block">\vartheta_{00}(x)^{1/6}\vartheta_{01}(x)^{-1/3}\biggl[\frac{\vartheta_{00}(x)^4 - \vartheta_{01}(x)^4}{16\,x}\biggr]^{1/24} = \sum_{n=0}^{\infty} Q(n)x^n = \prod_{k = 1}^{\infty} \frac{1}{1 - x^{2k - 1}}</math>