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{{Short description|Expression of a function as an infinite sum of simpler functions}}
{{More citations needed|date=August 2021}}{{Unreliable sources|date=March 2022}}

[[File:Taylor cos.gif|alt=Approximation of cosine by a Taylor series|thumb|An animation showing the [[Trigonometric functions|cosine function]] being approximated by successive truncations of its [[Maclaurin series]].]]
In [[mathematics]], a '''series expansion''' is a technique that expresses a [[Function (mathematics)|function]] as an infinite sum, or [[Series (mathematics)|series]], of simpler functions. It is a method for calculating a [[Function (mathematics)|function]] that cannot be expressed by just elementary operators (addition, subtraction, multiplication and division).<ref>{{Cite web|date=2013-11-07|title=Series and Expansions|url=https://math.libretexts.org/Bookshelves/Analysis/Supplemental_Modules_(Analysis)/Series_and_Expansions|access-date=2021-12-24|website=Mathematics LibreTexts|language=en}}</ref>

The resulting so-called ''[[Series (mathematics)|series]]'' often can be limited to a finite number of terms, thus yielding an [[approximation]] of the function. The fewer terms of the sequence are used, the simpler this approximation will be. Often, the resulting inaccuracy (i.e., the [[partial sum]] of the omitted terms) can be described by an equation involving [[Big O notation]] (see also [[asymptotic expansion]]). The series expansion on an [[open interval]] will also be an approximation for non-[[analytic functions]].<ref>{{Cite book|last1=Gil|first1=Amparo|url=https://books.google.com/books?id=WGHWBoX89doC&q=%22series+expansion%22+AND+%22approximation%22|title=Numerical Methods for Special Functions|last2=Segura|first2=Javier|last3=Temme|first3=Nico M.|date=2007-01-01|publisher=SIAM|isbn=978-0-89871-782-2|language=en}}</ref>{{Verify source|date=August 2021}}

== Types of series expansions ==
There are several kinds of series expansions, listed below.

=== Taylor series ===
A ''[[Taylor series]]'' is a [[power series]] based on a function's [[derivative]]s at a single point.<ref name=":1">{{Cite web |date=27 December 2013 |title=Taylor series - Encyclopedia of Mathematics |url=https://encyclopediaofmath.org/wiki/Taylor_series |access-date=22 March 2022 |website=encyclopediaofmath.org}}</ref> More specifically, if a function <math>f: U\to\R</math> is infinitely differentiable around a point <math>x_0</math>, then the Taylor series of ''f'' around this point is given by

<math>\sum_{n=0}^{\infty}\frac{f^{(n)}(x_0)}{n!}(x - x_0)^n</math>

under the convention <math>0^0 := 1</math>.<ref name=":1" /><ref name=":2">{{Cite book |last1=Edwards |first1=C. Henry |title=Elementary Differential Equations with Boundary Value Problems |last2=Penney |first2=David E. |year=2008 |isbn=978-0-13-600613-8 |pages=196 |language=en}}</ref> The ''[[Maclaurin series]]'' of ''f'' is its Taylor series about <math>x_0 = 0</math>.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Maclaurin Series |url=https://mathworld.wolfram.com/ |access-date=2022-03-22 |website=mathworld.wolfram.com |language=en}}</ref><ref name=":2" />

=== Laurent series ===
A ''[[Laurent series]]'' is a generalization of the Taylor series, allowing terms with negative exponents; it takes the form <math display="inline">\sum_{k = -\infty}^{\infty} c_k (z - a)^k</math> and converges in an [[Annulus (mathematics)|annulus]].<ref>{{Cite web |title=Laurent series - Encyclopedia of Mathematics |url=https://encyclopediaofmath.org/wiki/Laurent_series |access-date=2022-03-22 |website=encyclopediaofmath.org}}</ref> In particular, a Laurent series can be used to examine the behavior of a complex function near a singularity by considering the series expansion on an annulus centered at the singularity.

=== Dirichlet series ===
[[File:ZetaSpiral.gif|alt=spiral representing partial sums of the Dirichlet series defining the Riemann zeta function|thumb|Convergence and divergence of partial sums of the Dirichlet series defining the [[Riemann zeta function]]. Here, the yellow line represents the first fifty successive partial sums <math display="inline">\sum_{n = 1}^k n^{-s},</math> the magenta dotted line represents <math>\tfrac{n^{-s+1}}{-s+1} + \zeta(s),</math> and the green dot represents <math>\zeta(s)</math> as ''s'' is varied from -0.5 to 1.5.]]
A ''general [[Dirichlet series]]'' is a series of the form <math display="inline">\sum_{n = 1}^{\infty} a_ne^{-\lambda_n s}.</math> One important special case of this is the ''ordinary Dirichlet series'' <math display="inline">\sum_{n = 1}^{\infty}\frac{a_n}{n^s}.</math><ref name=":3">{{Cite web |date=26 January 2022 |title=Dirichlet series - Encyclopedia of Mathematics |url=https://encyclopediaofmath.org/wiki/Dirichlet_series |access-date=22 March 2022 |website=encyclopediaofmath.org}}</ref> Used in [[number theory]].{{Citation needed|date=August 2021}}

=== Fourier series ===

A ''[[Fourier series]]'' is an expansion of periodic functions as a sum of many [[Trigonometric functions#Sine, cosine, and tangent|sine]] and [[Trigonometric functions#Sine, cosine, and tangent|cosine]] functions.<ref name=":4">{{Cite web |title=Fourier series - Encyclopedia of Mathematics |url=https://encyclopediaofmath.org/wiki/Fourier_series |access-date=2022-03-22 |website=encyclopediaofmath.org}}</ref> More specifically, the Fourier series of a function <math>f(t)</math> of period <math>2L</math> is given by the expression<math display="block">a_0 + \sum_{n = 1}^{\infty} \left[a_n\cos\left(\frac{n\pi t}{L}\right) + b_n\sin\left(\frac{n\pi t}{L}\right)\right]</math>where the coefficients are given by the formulae<ref name=":4" /><ref>{{Cite book |last1=Edwards |first1=C. Henry |title=Elementary Differential Equations with Boundary Value Problems |last2=Penney |first2=David E. |year=2008 |isbn=978-0-13-600613-8 |pages=558, 564 |language=en}}</ref><math display="block">\begin{align}
a_n &:= \frac{1}{L}\int_{-L}^L f(t)\cos\left(\frac{n\pi t}{L}\right)dt, \\
b_n &:= \frac{1}{L}\int_{-L}^L f(t)\sin\left(\frac{n\pi t}{L}\right)dt.
\end{align}</math>

=== Other series ===
In [[acoustics]], e.g., the [[Fundamental frequency|fundamental tone]] and the [[overtone]]s together form an example of a Fourier series.{{Citation needed|date=March 2022}}

[[Table of Newtonian series|Newtonian series]]{{Citation needed|date=August 2021}}

[[Legendre polynomials]]: Used in [[physics]] to describe an arbitrary electrical field as a [[superposition principle|superposition]] of a [[dipole]] field, a [[quadrupole]] field, an [[Multipole expansion|octupole]] field, etc.{{Citation needed|date=August 2021}}

[[Zernike polynomials]]: Used in [[optics]] to calculate [[Optical aberration|aberration]]s of optical systems. Each term in the series describes a particular type of aberration.{{Citation needed|date=August 2021}}
[[File:Stirling series relative error.svg|thumb|The relative error in a truncated Stirling series vs. {{mvar|n}}, for 0 to 5 terms. The kinks in the curves represent points where the truncated series coincides with <math>\Gamma(n + 1).</math>]]
The ''[[Stirling series]]''<math display=block>\text{Ln}\Gamma\left(z\right)\sim\left(z-\tfrac{1}{2}\right)\ln z-z+\tfrac{1}{2}\ln\left(2\pi\right)+\sum_{k=1}^{\infty}\frac{B_{2k}}{2k(2k-1)z^{2k-1}}</math>is an approximation of the [[Gamma function#The log-gamma function|log-gamma function]].<ref>{{Cite web |title=DLMF: 5.11 Asymptotic Expansions |url=https://dlmf.nist.gov/5.11 |access-date=22 March 2022 |website=dlmf.nist.gov}}</ref>

==Examples==
The following is the [[Taylor series]] of <math>e^x</math>:<math display="block">e^x=\sum^{\infty}_{n=0}\frac{x^n}{n!}= 1 + x + \frac{x^2}{2} + \frac{x^3}{6}...</math><ref>{{Cite web|last=Weisstein|first=Eric W.|title=Exponential Function|url=https://mathworld.wolfram.com/ExponentialFunction.html|access-date=2021-08-12|website=mathworld.wolfram.com|language=en}}</ref><ref>{{Cite web|date=5 June 2020|title=Exponential function - Encyclopedia of Mathematics|url=https://encyclopediaofmath.org/wiki/Exponential_function|url-status=live|access-date=12 August 2021|website=encyclopediaofmath.org}}</ref>

The Dirichlet series of the [[Riemann zeta function]] is<math display="block">\zeta(s) := \sum_{n = 1}^{\infty} \frac{1}{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \cdots</math><ref name=":3" />

== References ==
<references />
[[Category:Series expansions| ]]
[[Category:Algebra]]
[[Category:Polynomials]]
[[Category:Mathematical analysis]]
[[Category:Mathematical series]]

Series expansion - Revision history

Ajay Kumar: Created a new article