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{{Short description|Mathematical approximation of a function}}
[[File:sintay_SVG.svg|thumb|300px|As the degree of the Taylor polynomial rises, it approaches the correct function. This image shows {{math|sin ''x''}} and its Taylor approximations by polynomials of degree <span style="color:red;">'''1'''</span>, <span style="color:orange;">'''3'''</span>, <span style="color:yellow;">'''5'''</span>, <span style="color:lime;">'''7'''</span>, <span style="color:blue;">'''9'''</span>, <span style="color:indigo;">'''11'''</span>, and <span style="color:violet;">'''13'''</span> at {{math|1=''x'' = 0}}.]]
{{Calculus |Series}}

In [[Mathematics|mathematics]], the '''Taylor series''' or '''Taylor expansion''' of a [[Function (mathematics)|function]] is an [[Series (mathematics)|infinite sum]] of terms that are expressed in terms of the function's [[Derivative|derivative]]s 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 [[Biography:Brook Taylor|Brook Taylor]], who introduced them in 1715. A Taylor series is also called a '''Maclaurin series''' when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century.

The partial sum formed by the first {{math|''n'' + 1}} terms of a Taylor series is a [[Polynomial|polynomial]] of degree {{mvar|n}} that is called the {{mvar|n}}th '''Taylor polynomial''' of the function. Taylor polynomials are approximations of a function, which become generally more accurate as {{mvar|n}} increases. [[Taylor's theorem]] gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the [[Limit of a sequence|limit]] of the infinite sequence 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 [[Analytic function|analytic]] at a point {{mvar|x}} if it is equal to the sum of its Taylor series in some open interval (or [[Disk (mathematics)|open disk]] in the [[Complex plane|complex plane]]) containing {{mvar|x}}. This implies that the function is analytic at every point of the interval (or disk).

==Definition==
The Taylor series of a [[Real-valued function|real]] or [[Complex-valued function|complex-valued function]] {{math|''f''&thinsp;(''x'')}}, that is infinitely differentiable at a [[Real number|real]] or [[Complex number|complex number]] {{math|''a''}}, is the [[Power series|power series]]
<!--
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<math display="block"> 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}. </math>
Here, {{math|''n''!}} denotes the [[Factorial|factorial]] of {{mvar|n}}. The function {{math|''f''{{isup|(''n'')}}(''a'')}} denotes the {{mvar|n}}th [[Derivative|derivative]] of {{mvar|f}} evaluated at the point {{mvar|a}}. The derivative of order zero of {{mvar|f}} is defined to be {{mvar|f}} itself and {{math|(''x'' − ''a'')<sup>0</sup>}} and {{math|0!}} [[Empty product|are both defined to be&nbsp;1]]. This series can be written by using sigma notation, as in the right side formula.{{sfn|Banner|2007|p=[https://books.google.com/books?id=OrumDwAAQBAJ&pg=PA530 530]}} With {{math|''a'' {{=}} 0}}, the Maclaurin series takes the form:{{sfn|Thomas|Finney|1996|loc=See §8.9.}}
<math display="block"> 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}. </math>

==Examples==
The Taylor series of any [[Polynomial|polynomial]] is the polynomial itself.

The Maclaurin series of {{math|{{sfrac|1|1 − ''x''}}}} is the [[Geometric series|geometric series]]

:<math>1 + x + x^2 + x^3 + \cdots.</math>

So, by substituting {{mvar|x}} for {{math|1 − ''x''}}, the Taylor series of {{math|{{sfrac|1|''x''}}}} at {{math|''a'' {{=}} 1}} is

:<math>1 - (x-1) + (x-1)^2 - (x-1)^3 + \cdots.</math>

By integrating the above Maclaurin series, we find the Maclaurin series of {{math|ln(1  − ''x'')}}, where {{math|ln}} denotes the [[Natural logarithm|natural logarithm]]:

:<math>-x - \tfrac{1}{2}x^2 - \tfrac{1}{3}x^3 - \tfrac{1}{4}x^4 - \cdots.</math>

The corresponding Taylor series of {{math|ln ''x''}} at {{math|''a'' {{=}} 1}} is

:<math>(x-1) - \tfrac{1}{2}(x-1)^2 + \tfrac{1}{3}(x-1)^3 - \tfrac{1}{4}(x-1)^4 + \cdots,</math>

and more generally, the corresponding Taylor series of {{math|ln ''x''}} at an arbitrary nonzero point {{mvar|a}} is:

:<math>\ln a + \frac{1}{a} (x - a) - \frac{1}{a^2}\frac{\left(x - a\right)^2}{2} + \cdots.</math>

The Maclaurin series of the [[Exponential function|exponential function]] {{math|''e''<sup>''x''</sup>}} is

:<math>\begin{align}
   \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{align}</math>

The above expansion holds because the derivative of {{math|''e''<sup>''x''</sup>}} with respect to {{mvar|x}} is also {{math|''e''<sup>''x''</sup>}}, and {{math|''e''<sup>0</sup>}} equals&nbsp;1. This leaves the terms {{math|(''x'' − 0)<sup>''n''</sup>}} in the numerator and {{math|''n''!}} in the denominator of each term in the infinite sum.

== History ==
The ancient Greek philosopher [[Biography:Zeno of Elea|Zeno of Elea]] considered the problem of summing an infinite series to achieve a finite result, but rejected it as an impossibility;{{sfn|Lindberg|2007|p=33}} the result was Zeno's paradox. Later, [[Biography:Aristotle|Aristotle]] proposed a philosophical resolution of the paradox, but the mathematical content was apparently unresolved until taken up by [[Biography:Archimedes|Archimedes]], as it had been prior to Aristotle by the Presocratic Atomist [[Biography:Democritus|Democritus]]. It was through Archimedes's [[Method of exhaustion|method of exhaustion]] that an infinite number of progressive subdivisions could be performed to achieve a finite result.{{sfn|Kline|1990|p=[https://archive.org/details/mathematicalthou00klin/page/n437 35]–37}} [[Biography:Liu Hui|Liu Hui]] independently employed a similar method a few centuries later.{{sfn|Boyer|Merzbach|1991|p=[https://archive.org/details/historyofmathema00boye/page/202 202–203]}}<!--I'm sure there are better refs than this. Hui gave fairly "rigorous" bounds on the convergence, if I recall. But it isn't addressed here.-->

In the 14th century, the earliest examples of specific Taylor series (but not the general method) were given by [[Biography:Madhava of Sangamagrama|Madhava of Sangamagrama]].{{sfn|Dani|2012}} Though no record of his work survives, writings of his followers in the [[Astronomy:Kerala school of astronomy and mathematics|Kerala school of astronomy and mathematics]] suggest that he found the Taylor series for the trigonometric functions of [[Sine|sine]], [[Cosine|cosine]], and arctangent (see [[Madhava series]]). During the following two centuries his followers developed further series expansions and rational approximations.

In late 1670, [[Biography:James Gregory (mathematician)|James Gregory]] was shown in a letter from John Collins several Maclaurin series {{nobr|(<math display=inline>\sin x,</math>}} <math display=inline>\cos x,</math> <math display=inline>\arcsin x,</math> and {{nobr|<math display=inline>x \cot x</math>)}} derived by [[Biography:Isaac Newton|Isaac Newton]], 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 <math display=inline>\arctan x,</math> <math display=inline>\tan x,</math> <math display=inline>\sec x,</math> <math display=inline>\ln\, \sec x</math> (the integral of {{nobr|<math>\tan</math>),}} <math display=inline>\ln\, \tan\tfrac12{\bigl(\tfrac12\pi + x\bigr)}</math> (the [[Integral of the secant function|integral of {{math|sec}}]], the inverse [[Gudermannian function]]), <math display=inline>\arcsec \bigl(\sqrt2 e^x\bigr),</math> and <math display=inline>2 \arctan e^x - \tfrac12\pi</math> (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.<ref>{{multiref|{{harvnb|Turnbull|1939|pp=168–174}}|{{harvnb|Roy|1990}}|{{harvnb|Malet|1993}}}}</ref>

In 1691–1692, Isaac Newton wrote down an explicit statement of the Taylor and Maclaurin series in an unpublished version of his work ''De Quadratura Curvarum''. However, this work was never completed and the relevant sections were omitted from the portions published in 1704 under the title ''Tractatus de Quadratura Curvarum''.

It was not until 1715 that a general method for constructing these series for all functions for which they exist was finally published by [[Biography:Brook Taylor|Brook Taylor]], <ref>{{multiref|{{harvnb|Taylor|1715|p=21–23, see Prop. VII, Thm. 3, Cor. 2}}. See {{harvnb|Struik|1969|pp=329–332}} for English translation, and {{harvnb|Bruce|2007}} for re-translation.|{{harvnb|Feigenbaum|1985}}}}</ref> after whom the series are now named.

The Maclaurin series was named after Colin Maclaurin, a professor in Edinburgh, who published the special case of the Taylor result in the mid-18th century.

==Analytic functions==
[[Image:Exp neg inverse square.svg|300px|thumb|right|The function {{math|1=<strong style="color:#803300">''e''<sup>(−1/''x''<sup>2</sup>)</sup></strong>}} is not analytic at {{math|1=''x'' {{=}} 0}}: the Taylor series is identically 0, although the function is not.]]
{{main|Analytic function}}
If {{math|''f''&thinsp;(''x'')}} is given by a convergent power series in an open disk centred at {{mvar|b}} in the complex plane (or an interval in the real line), it is said to be [[Analytic function|analytic]] in this region.  Thus for {{mvar|x}} in this region, {{mvar|f}} is given by a convergent power series
:<math>f(x) = \sum_{n=0}^\infty a_n(x-b)^n.</math>
Differentiating by {{mvar|x}} the above formula {{mvar|n}} times, then setting {{math|''x'' {{=}} ''b''}} gives:
:<math>\frac{f^{(n)}(b)}{n!} = a_n</math>
and so the power series expansion agrees with the Taylor series.  Thus a function is analytic in an open disk centered at {{mvar|b}} if and only if its Taylor series converges to the value of the function at each point of the disk.

If {{math|''f''&thinsp;(''x'')}} is equal to the sum of its Taylor series for all {{mvar|x}} in the complex plane, it is called [[Entire function|entire]]. The polynomials, [[Exponential function|exponential function]] {{math|''e''<sup>''x''</sup>}}, and the trigonometric functions sine and cosine, are examples of entire functions. Examples of functions that are not entire include the [[Square root|square root]], the [[Logarithm|logarithm]], the trigonometric function tangent, and its inverse, arctan. For these functions the Taylor series do not [[Convergent series|converge]] if {{mvar|x}} is far from {{mvar|b}}. That is, the Taylor series [[Divergent series|diverges]] at {{mvar|x}} if the distance between {{mvar|x}} and {{mvar|b}} is larger than the [[Radius of convergence|radius of convergence]]. 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.

Uses of the Taylor series for analytic functions include:
# The partial sums (the Taylor polynomials) of the series can be used as approximations of the function. These approximations are good if sufficiently many terms are included.
#Differentiation and integration of power series can be performed term by term and is hence particularly easy.
#An [[Analytic function|analytic function]] is uniquely extended to a [[Holomorphic function|holomorphic function]] on an open disk in the [[Complex number|complex plane]]. This makes the machinery of [[Complex analysis|complex analysis]] available.
#The (truncated) series can be used to compute function values numerically, (often by recasting the polynomial into the Chebyshev form and evaluating it with the [[Clenshaw algorithm]]).
#Algebraic operations can be done readily on the power series representation; for instance, [[Euler's formula]] follows from Taylor series expansions for trigonometric and exponential functions. This result is of fundamental importance in such fields as [[Physics:Harmonic analysis|harmonic analysis]].
#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.

==Approximation error and convergence==
{{main|Taylor's theorem}}
[[Image:Taylorsine.svg|300px|thumb|right|The sine function (blue) is closely approximated by its Taylor polynomial of degree 7 (pink) for a full period centered at the origin.]]
[[Image:LogTay.svg|300px|thumb|right|The Taylor polynomials for {{math|ln(1 + ''x'')}} only provide accurate approximations in the range {{math|−1 < ''x'' ≤ 1}}. For {{math|''x'' > 1}}, Taylor polynomials of higher degree provide worse approximations.]]
[[Image:Logarithm GIF.gif|300px|thumb|right|The Taylor approximations for {{math|ln(1 + ''x'')}} (black). For {{math|''x'' > 1}}, the approximations diverge.]]

Pictured is an accurate approximation of {{math|sin ''x''}} around the point {{math|''x'' {{=}} 0}}. The pink curve is a polynomial of degree seven:

:<math>\sin{x} \approx x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!}.\!</math>

The error in this approximation is no more than {{math|{{abs|''x''}}<sup>9</sup>&thinsp;/&thinsp;9!}}. For a full cycle centered at the origin ({{math|−π < ''x'' < π}}) the error is less than 0.08215. In particular, for {{math|−1 < ''x'' < 1}}, the error is less than&nbsp;0.000003.

In contrast, also shown is a picture of the natural logarithm function {{math|ln(1 + ''x'')}} and some of its Taylor polynomials around {{math|''a'' {{=}} 0}}. These approximations converge to the function only in the region {{math|−1 < ''x'' ≤ 1}}; outside of this region the higher-degree Taylor polynomials are ''worse'' approximations for the function.

The ''error'' incurred in approximating a function by its {{mvar|n}}th-degree Taylor polynomial is called the ''remainder'' or ''[[Residual (numerical analysis)|residual]]'' and is denoted by the function {{math|''R''<sub>''n''</sub>(''x'')}}. Taylor's theorem can be used to obtain a bound on the [[Taylor's theorem#Explicit formulas for the remainder|size of the remainder]].

In general, Taylor series need not be [[Convergent series|convergent]] at all. And in fact the set of functions with a convergent Taylor series is a meager set in the [[Fréchet space]] of smooth functions. And even if the Taylor series of a function {{mvar|f}} does converge, its limit need not in general be equal to the value of the function {{math|''f''&thinsp;(''x'')}}. For example, the function
:<math>
f(x) = \begin{cases}
  e^{-1/x^2} & \text{if } x \neq 0 \\[3mu]
  0          & \text{if } x = 0
\end{cases}
</math>
is infinitely differentiable at {{math|''x'' {{=}} 0}}, and has all derivatives zero there. Consequently, the Taylor series of {{math|''f''&thinsp;(''x'')}} about {{math|''x'' {{=}} 0}} is identically zero. However, {{math|''f''&thinsp;(''x'')}} is not the zero function, so does not equal its Taylor series around the origin.  Thus, {{math|''f''&thinsp;(''x'')}} is an example of a [[Non-analytic smooth function|non-analytic smooth function]].

In [[Real analysis|real analysis]], this example shows that there are infinitely differentiable functions {{math|''f''&thinsp;(''x'')}} whose Taylor series are ''not'' equal to {{math|''f''&thinsp;(''x'')}} even if they converge. By contrast, the [[Holomorphic function|holomorphic function]]s studied in [[Complex analysis|complex analysis]] always possess a convergent Taylor series, and even the Taylor series of [[Meromorphic function|meromorphic function]]s, which might have singularities, never converge to a value different from the function itself. The complex function {{math|''e''<sup>−1/''z''<sup>2</sup></sup>}}, however, does not approach 0 when {{mvar|z}} approaches 0 along the imaginary axis, so it is not [[Continuous function|continuous]] in the complex plane and its Taylor series is undefined at 0.

More generally, every sequence of real or complex numbers can appear as [[Coefficient|coefficient]]s in the Taylor series of an infinitely differentiable function defined on the real line, a consequence of [[Borel's lemma]].  As a result, the [[Radius of convergence|radius of convergence]] 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.{{sfn|Rudin|1980|p=418|loc=See Exercise 13}}

A function cannot be written as a Taylor series centred at a [[Singularity (mathematics)|singularity]]; in these cases, one can often still achieve a series expansion if one allows also negative powers of the variable {{mvar|x}}; see [[Laurent series]]. For example, {{math|''f''&thinsp;(''x'') {{=}} ''e''<sup>−1/''x''<sup>2</sup></sup>}} can be written as a Laurent series.

===Generalization===
The generalization of the Taylor series does converge to the value of the function itself for any [[Bounded function|bounded]] [[Continuous function|continuous function]] on {{math|(0,∞)}}, and this can be done by using the calculus of finite differences. Specifically, the following theorem, due to [[Biography:Einar Hille|Einar Hille]], that for any {{math|''t'' > 0}},<ref>{{multiref
 |{{harvnb|Feller|2003|p=230&ndash;232}}
 |{{harvnb|Hille|Phillips|1957|pp=300–327}}
}}</ref>
<math display="block" >\lim_{h\to 0^+}\sum_{n=0}^\infty \frac{t^n}{n!}\frac{\Delta_h^nf(a)}{h^n} = f(a+t).</math>
Here {{math|Δ{{su|p=''n''|b=''h''}}}} is the {{mvar|n}}th finite difference operator with step size {{mvar|h}}. The series is precisely the Taylor series, except that divided differences appear in place of differentiation: the series is formally similar to the Newton series. When the function {{mvar|f}} is analytic at {{mvar|a}}, the terms in the series converge to the terms of the Taylor series, and in this sense generalizes the usual Taylor series.

In general, for any infinite sequence {{math|''a''<sub>''i''</sub>}}, the following power series identity holds:
:<math>\sum_{n=0}^\infty\frac{u^n}{n!}\Delta^na_i = e^{-u}\sum_{j=0}^\infty\frac{u^j}{j!}a_{i+j}.</math>
So in particular,
:<math>f(a+t) = \lim_{h\to 0^+} e^{-t/h}\sum_{j=0}^\infty f(a+jh) \frac{(t/h)^j}{j!}.</math>
The series on the right is the [[Expected value|expected value]] of {{math|''f''&thinsp;(''a'' + ''X'')}}, where {{mvar|X}} is a [[Poisson distribution|Poisson-distributed]] [[Random variable|random variable]] that takes the value {{math|''jh''}} with probability {{math|''e''<sup>−''t''/''h''</sup>·{{sfrac|(''t''/''h''){{isup|''j''}}|''j''!}}}}. Hence,
:<math>f(a+t) = \lim_{h\to 0^+} \int_{-\infty}^\infty f(a+x)dP_{t/h,h}(x).</math>
The [[Law of large numbers|law of large numbers]] implies that the identity holds.{{sfn|Feller|2003|p=231}}

==List of Maclaurin series of some common functions==
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|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|exponential function]] <math>e^x</math> (with base {{mvar|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|generating function]] of the [[Bell number|Bell numbers]] 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|natural logarithm]] (with base {{mvar|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 [[Biography:Nicholas Mercator|Nicholas Mercator]] in his ''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|geometric series]] and its derivatives have Maclaurin series

:<math>\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 given in the next section.

=== Binomial series ===
The [[Binomial series|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|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|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|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|binomial approximation]].

=== Trigonometric functions ===
The usual trigonometric functions 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>\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|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|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>\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|polylogarithms]] have these defining identities:
:<math>\text{Li}_{2}(x) = \sum_{n = 1}^{\infty} \frac{1}{n^2} x^{n}</math>
:<math>\text{Li}_{3}(x) = \sum_{n = 1}^{\infty} \frac{1}{n^3} x^{n}</math>

The [[Legendre chi function|Legendre chi functions]] are defined as follows:
:<math>\chi_{2}(x) = \sum_{n = 0}^{\infty} \frac{1}{(2n + 1)^2} x^{2n + 1}</math>
:<math>\chi_{3}(x) = \sum_{n = 0}^{\infty} \frac{1}{(2n + 1)^3} x^{2n + 1}</math>

And the formulas presented below are called ''[[Inverse tangent integral|inverse tangent integrals]]'':
:<math>\text{Ti}_{2}(x) = \sum_{n = 0}^{\infty} \frac{(-1)^{n}}{(2n + 1)^2} x^{2n + 1}</math>
:<math>\text{Ti}_{3}(x) = \sum_{n = 0}^{\infty} \frac{(-1)^{n}}{(2n + 1)^3} x^{2n + 1}</math>

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

=== Elliptic functions ===

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

:<math>\frac{2}{\pi}K(x) = \sum_{n = 0}^{\infty} \frac{[(2n)!]^2}{16^{n}(n!)^4}x^{2n}</math>
:<math>\frac{2}{\pi}E(x) = \sum_{n = 0}^{\infty} \frac{[(2n)!]^2}{(1 - 2n)16^{n}(n!)^4}x^{2n}</math>

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

:<math>\vartheta_{00}(x) = 1 + 2\sum_{n = 1}^{\infty} x^{n^2}</math>
:<math>\vartheta_{01}(x) = 1 + 2\sum_{n = 1}^{\infty} (-1)^{n} x^{n^2}</math>

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

:<math>\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>\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>

==Calculation of Taylor series==
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 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 [[Integration by parts|integration by parts]]. Particularly convenient is the use of [[Software:Computer algebra system|computer algebra system]]s to calculate Taylor series.

===First example===
In order to compute the 7th degree Maclaurin polynomial for the function
:<math>f(x)=\ln(\cos x),\quad x\in\bigl({-\tfrac\pi2}, \tfrac\pi2\bigr),</math>
one may first rewrite the function as
:<math>f(x)={\ln}\bigl(1+(\cos x-1)\bigr),</math>
the composition of two functions <math>x \mapsto \ln(1 + x)</math> and <math>x \mapsto \cos x - 1.</math> The Taylor series for the natural logarithm is (using [[Big O notation|big O notation]])
:<math>\ln(1+x) = x - \frac{x^2}2 + \frac{x^3}3 + O{\left(x^4\right)}</math>
and for the cosine function
:<math>\cos x - 1 = -\frac{x^2}2 + \frac{x^4}{24} - \frac{x^6}{720} + O{\left(x^8\right)}.</math>
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:

:<math>\begin{align}f(x)&=\ln\bigl(1+(\cos x-1)\bigr)\\
&=(\cos x-1) - \tfrac12(\cos x-1)^2 + \tfrac13(\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{align}\!</math>
Since the cosine is an even function, the coefficients for all the odd powers are zero.

===Second example===
Suppose we want the Taylor series at 0 of the function
: <math>g(x)=\frac{e^x}{\cos x}.\!</math>
The Taylor series for the exponential function is
: <math>e^x =1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!}+\cdots,</math>
and the series for cosine is
: <math>\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots.</math>
Assume the series for their quotient is
: <math>\frac{e^x}{\cos x} = c_0 + c_1 x + c_2 x^2 + c_3 x^3 + c_4x^4 + \cdots</math>
Multiplying both sides by the denominator <math>\cos x</math> and then expanding it as a series yields
: <math>\begin{align}
e^x 
&=\left(c_0 + c_1 x + c_2 x^2 + c_3 x^3 + c_4x^4 + \cdots\right)\left(1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots\right)\\[5mu]
&=c_0 + c_1x + \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{align}</math>
Comparing the coefficients of <math>g(x)\cos x</math> with the coefficients of <math>e^x,</math>

:<math>
c_0 = 1,\ \ c_1 = 1,\ \ 
c_2 - \tfrac12 c_0 = \tfrac12,\ \ c_3 - \tfrac12 c_1 = \tfrac16,\ \ 
c_4 - \tfrac12 c_2 + \tfrac1{24} c_0 = \tfrac1{24},\ \ldots.
</math>

The coefficients <math>c_i</math> of the series for <math>g(x)</math> can thus be computed one at a time, amounting to long division of the series for <math>e^x</math> and {{nobr|<math>\cos x</math>:}}
: <math>\frac{e^x}{\cos x}=1 + x + x^2 + \tfrac23 x^3 + \tfrac12 x^4 + \cdots.</math>

===Third example===
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 {{math|(1 + ''x'')''e<sup>x</sup>''}} as a Taylor series in {{mvar|x}}, we use the known Taylor series of function {{math|''e''<sup>''x''</sup>}}:
: <math>e^x = \sum^\infty_{n=0} \frac{x^n}{n!} =1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!}+\cdots.</math>
Thus,
: <math>\begin{align}(1+x)e^x &= e^x + xe^x = \sum^\infty_{n=0} \frac{x^n}{n!} + \sum^\infty_{n=0} \frac{x^{n+1}}{n!} = 1 + \sum^\infty_{n=1} \frac{x^n}{n!} + \sum^\infty_{n=0} \frac{x^{n+1}}{n!} \\ &= 1 + \sum^\infty_{n=1} \frac{x^n}{n!} + \sum^\infty_{n=1} \frac{x^n}{(n-1)!} =1 + \sum^\infty_{n=1}\left(\frac{1}{n!} + \frac{1}{(n-1)!}\right)x^n \\ &= 1 + \sum^\infty_{n=1}\frac{n+1}{n!}x^n\\ &= \sum^\infty_{n=0}\frac{n+1}{n!}x^n.\end{align}</math>

==Taylor series as definitions==
Classically, [[Algebraic function|algebraic function]]s are defined by an algebraic equation, and [[Transcendental function|transcendental function]]s (including those discussed above) are defined by some property that holds for them, such as a [[Differential equation|differential equation]]. For example, the [[Exponential function|exponential function]] 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 [[Analytic function|analytic function]] by its Taylor series.

Taylor series are used to define functions and "[[Operator (mathematics)|operator]]s" 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 [[Matrix exponential|matrix exponential]] or matrix logarithm.

In other areas, such as formal analysis, it is more convenient to work directly with the [[Power series|power series]] themselves. Thus one may define a solution of a differential equation ''as'' a power series which, one hopes to prove, is the Taylor series of the desired solution.

==Taylor series in several variables{{anchor|In several variables}}==

The Taylor series may also be generalized to functions of more than one variable with<ref>{{multiref|{{harvnb|Hörmander|2002|loc=See Eqq. 1.1.7 and 1.1.7′}}
 |{{harvnb|Kolk|Duistermaat|2010|p=59–63}}
}}</ref>
:<math>\begin{align}
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{align}</math>

For example, for a function <math>f(x,y)</math> that depends on two variables, {{mvar|x}} and {{mvar|y}}, the Taylor series to second order about the point {{math|(''a'', ''b'')}} is

:<math>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)</math>

where the subscripts denote the respective [[Partial derivative|partial derivative]]s.

===Second-order Taylor series in several variables===

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

:<math>T(\mathbf{x}) = f(\mathbf{a}) + (\mathbf{x} - \mathbf{a})^\mathsf{T} D f(\mathbf{a}) + \frac{1}{2!} (\mathbf{x} - \mathbf{a})^\mathsf{T} \left \{D^2 f(\mathbf{a}) \right \} (\mathbf{x} - \mathbf{a}) + \cdots,</math>

where {{math|''D'' ''f''&thinsp;('''a''')}} is the [[Gradient|gradient]] of {{mvar|f}} evaluated at {{math|'''x''' {{=}} '''a'''}} and {{math|''D''<sup>2</sup> ''f''&thinsp;('''a''')}} is the [[Hessian matrix]]. Applying the [[Multi-index notation|multi-index notation]] the Taylor series for several variables becomes

:<math>T(\mathbf{x}) = \sum_{|\alpha| \geq 0}\frac{(\mathbf{x}-\mathbf{a})^\alpha}{\alpha !} \left({\mathrm{\partial}^{\alpha}}f\right)(\mathbf{a}),</math>

which is to be understood as a still more abbreviated multi-index version of the first equation of this paragraph, with a full analogy to the single variable case.

=== Example ===
[[Image:Second Order Taylor.svg|200px|thumb|right|Second-order Taylor series approximation (in orange) of a function {{math|''f''&thinsp;(''x'',''y'') {{=}} ''e<sup>x</sup>'' ln(1 + ''y'')}} around the origin.]]
In order to compute a second-order Taylor series expansion around point {{math|(''a'', ''b'') {{=}} (0, 0)}} of the function
:<math>f(x,y)=e^x\ln(1+y),</math>

one first computes all the necessary partial derivatives:

:<math>\begin{align}
f_x &= e^x\ln(1+y) \\[6pt]
f_y &= \frac{e^x}{1+y} \\[6pt]
f_{xx} &= e^x\ln(1+y) \\[6pt]
f_{yy}  &= - \frac{e^x}{(1+y)^2}  \\[6pt]
f_{xy} &=f_{yx} =  \frac{e^x}{1+y} .
\end{align}</math>

Evaluating these derivatives at the origin gives the Taylor coefficients

:<math>\begin{align}
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{align}</math>

Substituting these values in to the general formula

:<math>\begin{align}
T(x,y) = &f(a,b) +(x-a) f_x(a,b) +(y-b) f_y(a,b) 
\\ &{}+\frac{1}{2!}\left( (x-a)^2f_{xx}(a,b) 
+ 2(x-a)(y-b)f_{xy}(a,b) +(y-b)^2 f_{yy}(a,b) \right)+ \cdots
\end{align}</math>

produces

:<math>\begin{align}
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 - \tfrac12 y^2 + \cdots 
\end{align}</math>

Since {{math|ln(1 + ''y'')}} is analytic in {{math|{{abs|''y''}} < 1}}, we have
:<math>e^x\ln(1+y)= y + xy - \tfrac12 y^2 + \cdots, \qquad |y| < 1.</math>

== Comparison with Fourier series ==
{{main|Fourier series}}
The trigonometric [[Fourier series]] enables one to express a [[Periodic function|periodic function]] (or a function defined on a closed interval {{math|[''a'',''b'']}}) as an infinite sum of trigonometric functions ([[Sine|sine]]s and [[Cosine|cosine]]s). 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 powers. Nevertheless, the two series differ from each other in several relevant issues:
* The finite truncations of the Taylor series of {{math|''f''&thinsp;(''x'')}} about the point {{math|''x'' {{=}} ''a''}} are all exactly equal to {{math|''f''}} at {{math|''a''}}. 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.
* The computation of Taylor series requires the knowledge of the function on an arbitrary small [[Neighbourhood (mathematics)|neighbourhood]] of a point, whereas the computation of the Fourier series requires knowing the function on its whole domain [[Interval (mathematics)|interval]]. In a certain sense one could say that the Taylor series is "local" and the Fourier series is "global".
* 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 integrable function.  In particular, the function could be nowhere differentiable. (For example, {{math|''f''&thinsp;(''x'')}} could be a [[Weierstrass function]].)
* 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 [[Pointwise convergence|pointwise]] to the function, and [[Uniform convergence|uniformly]] on every compact subset of the convergence interval. Concerning the Fourier series, if the function is [[Square-integrable function|square-integrable]] then the series converges in quadratic mean, but additional requirements are needed to ensure the pointwise or uniform convergence (for instance, if the function is periodic and of class C<sup>1</sup> then the convergence is uniform).
* 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 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.

==See also==
* [[Asymptotic expansion]]
* [[Generating function]]
* [[Laurent series]]
* [[Madhava series]]
* [[Newton polynomial|Newton's divided difference interpolation]]
* [[Padé approximant]]
* [[Puiseux series]]
* [[Shift operator]]

==Notes==
{{Reflist|30em}}

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* {{cite book|language = la
 | last = Taylor | first = Brook
 | title = Methodus Incrementorum Directa et Inversa
 | url = https://archive.org/details/UFIE003454_TO0324_PNI-2529_000000
 | trans-title = Direct and Reverse Methods of Incrementation
 | location = London
 | date = 1715
}}
* {{cite book
 | last1 = Thomas | first1 = George B. Jr.
 | last2 = Finney | first2 = Ross L.
 | title = Calculus and Analytic Geometry
 | edition = 9th
 | publisher = Addison Wesley
 | year = 1996
 | isbn = 0-201-53174-7
}}
* {{cite book
 | editor-last = Turnbull | editor-first = Herbert Westren
 | year = 1939
 | title = James Gregory; Tercentenary Memorial Volume
 | publisher = G. Bell & Sons
}}
{{refend}}

==External links==
{{Sister project links
|wikt=Taylor series
|commons=Category:Taylor series
|b=Calculus/Taylor series
|v=Taylor's series
|n=no |q=no |s=no |species=no
|d=Q131187}}
* {{springer|title=Taylor series|id=p/t092320}}
* {{MathWorld| urlname= TaylorSeries| title= Taylor Series}}
{{series (mathematics)}}

[[Category:Real analysis]]
[[Category:Complex analysis]]
[[Category:Series expansions]]

{{Sourceattribution|Taylor series}}