In mathematics, a power series (in one variable) is an infinite series of the form$$\backslash sum\_^\backslash infty\; a\_n\; \backslash left(x\; \; c\backslash right)^n\; =\; a\_0\; +\; a\_1\; (x\; \; c)\; +\; a\_2\; (x\; \; c)^2\; +\; \backslash cdots$$where a_{n} represents the coefficient of the nth term and c is a constant. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function.
In many situations c (the center of the series) is equal to zero, for instance when considering a Maclaurin series. In such cases, the power series takes the simpler form$$\backslash sum\_^\backslash infty\; a\_n\; x^n\; =\; a\_0\; +\; a\_1\; x\; +\; a\_2\; x^2\; +\; \backslash cdots.$$
Beyond their role in mathematical analysis, power series also occur in combinatorics as generating functions (a kind of formal power series) and in electronic engineering (under the name of the Ztransform). The familiar decimal notation for real numbers can also be viewed as an example of a power series, with integer coefficients, but with the argument x fixed at . In number theory, the concept of padic numbers is also closely related to that of a power series.
Any polynomial can be easily expressed as a power series around any center c, although all but finitely many of the coefficients will be zero since a power series has infinitely many terms by definition. For instance, the polynomial $f(x)\; =\; x^2\; +\; 2x\; +\; 3$ can be written as a power series around the center $c\; =\; 0$ as
f(x)=3+2x+1x^{2}+0x^{3}+0x^{4}+ …
f(x)=6+4(x1)+1(x1)^{2}+0(x1)^{3}+0(x1)^{4}+ …
or indeed around any other center c.^{[1]} One can view power series as being like "polynomials of infinite degree," although power series are not polynomials.
The geometric series formula
1  
1x 
=
infty  
\sum  
n=0 
x^{n}=1+x+x^{2}+x^{3}+ … ,
which is valid for $x\; <\; 1$, is one of the most important examples of a power series, as are the exponential function formula
e^{x}=
infty  
\sum  
n=0 
x^{n}  
n! 
=1+x+
x^{2}  
2! 
+
x^{3}  
3! 
+ … ,
and the sine formula
\sin(x)=
infty  
\sum  
n=0 
(1)^{n}x^{2n+1}  
(2n+1)! 
=x
x^{3}  
3! 
+
x^{5}  
5! 

x^{7}  
7! 
+ … ,
valid for all real x.
These power series are also examples of Taylor series.
Negative powers are not permitted in a power series; for instance, $1\; +\; x^\; +\; x^\; +\; \backslash cdots$ is not considered a power series (although it is a Laurent series). Similarly, fractional powers such as $x^\backslash frac$ are not permitted (but see Puiseux series). The coefficients $a\_n$ are not allowed to depend on thus for instance:
\sin(x)x+\sin(2x)x^{2}+\sin(3x)x^{3}+ …
is not a power series.
A power series
infty  
style\sum  
n=0 
n  
a  
n(xc) 
(xc)^{0}
a_{0}
r=\liminf_{n\toinfty}
 
\lefta  
n\right 
or, equivalently,
r^{1}=\limsup_{n\toinfty}
 
\lefta  
n\right 
(this is the Cauchy–Hadamard theorem; see limit superior and limit inferior for an explanation of the notation). The relation
r^{1}=\lim_{n\toinfty}\left{a_{n+1}\overa_{n}\right}
The set of the complex numbers such that is called the disc of convergence of the series. The series converges absolutely inside its disc of convergence, and converges uniformly on every compact subset of the disc of convergence.
For, there is no general statement on the convergence of the series. However, Abel's theorem states that if the series is convergent for some value such that, then the sum of the series for is the limit of the sum of the series for where is a real variable less than that tends to .
When two functions f and g are decomposed into power series around the same center c, the power series of the sum or difference of the functions can be obtained by termwise addition and subtraction. That is, if
f(x)=
infty  
\sum  
n=0 
a_{n}(xc)^{n}
g(x)=
infty  
\sum  
n=0 
b_{n}(xc)^{n}
then
f(x)\pmg(x)=
infty  
\sum  
n=0 
(a_{n}\pmb_{n)}(xc)^{n.}
It is not true that if two power series $\backslash sum\_^\backslash infty\; a\_n\; x^n$ and $\backslash sum\_^\backslash infty\; b\_n\; x^n$ have the same radius of convergence, then $\backslash sum\_^\backslash infty\; \backslash left(a\_n\; +\; b\_n\backslash right)\; x^n$ also has this radius of convergence. If $a\_n\; =\; (1)^n$ and $b\_n\; =\; (1)^\; \backslash left(1\; \; \backslash frac\backslash right)$, then both series have the same radius of convergence of 1, but the series $\backslash sum\_^\backslash infty\; \backslash left(a\_n\; +\; b\_n\backslash right)\; x^n\; =\; \backslash sum\_^\backslash infty\; \backslash frac\; x^n$ has a radius of convergence of 3.
With the same definitions for
f(x)
g(x)
\begin{align} f(x)g(x)&=
infty  
\left(\sum  
n=0 
a_{n}
infty  
(xc)  
n=0 
b_{n}(xc)^{n\right)}\\ &=
infty  
\sum  
i=0 
infty  
\sum  
j=0 
a_{i}b_{j}(xc)^{i+j}\\ &=
infty  
\sum  
n=0 
n  
\left(\sum  
i=0 
a_{i}b_{ni}\right)(xc)^{n. \end{align}}
The sequence $m\_n\; =\; \backslash sum\_^n\; a\_i\; b\_$ is known as the convolution of the sequences
a_{n}
For division, if one defines the sequence
d_{n}
f(x)  
g(x) 
=
 

=
infty  
\sum  
n=0 
d_{n}(xc)^{n}
then
f(x)=
infty  
\left(\sum  
n=0 
b_{n}(x
infty  
c)  
n=0 
d_{n}(xc)^{n\right)}
and one can solve recursively for the terms
d_{n}
Solving the corresponding equations yields the formulae based on determinants of certain matrices of the coefficients of
f(x)
g(x)
d  

d  

\begin{vmatrix} a_{n}&b_{1}&b_{2}& … &b_{n}\\ a_{n1}&b_{0}&b_{1}& … &b_{n1}\\ a_{n2}&0&b_{0}& … &b_{n2}\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ a_{0}&0&0& … &b_{0\end{vmatrix}}
Once a function
f(x)
\begin{align} f'(x)&=
infty  
\sum  
n=1 
a_{n}n(xc)^{n1}=
infty  
\sum  
n=0 
a_{n+1}(n+1)(xc)^{n,}\\ \intf(x)dx&=
infty  
\sum  
n=0 
a_{n}(xc)^{n+1}  
n+1 
+k=
infty  
\sum  
n=1 
a_{n1}(xc)^{n}  
n 
+k. \end{align}
Both of these series have the same radius of convergence as the original one.
See main article: Analytic function. A function f defined on some open subset U of R or C is called analytic if it is locally given by a convergent power series. This means that every a ∈ U has an open neighborhood V ⊆ U, such that there exists a power series with center a that converges to f(x) for every x ∈ V.
Every power series with a positive radius of convergence is analytic on the interior of its region of convergence. All holomorphic functions are complexanalytic. Sums and products of analytic functions are analytic, as are quotients as long as the denominator is nonzero.
If a function is analytic, then it is infinitely differentiable, but in the real case the converse is not generally true. For an analytic function, the coefficients a_{n} can be computed as
a_{n}=
f^{\left(}\left(c\right)  
n! 
where
f^{(n)}(c)
f^{(0)}(c)=f(c)
The global form of an analytic function is completely determined by its local behavior in the following sense: if f and g are two analytic functions defined on the same connected open set U, and if there exists an element c∈U such that f(c) = g(c) for all n ≥ 0, then f(x) = g(x) for all x ∈ U.
If a power series with radius of convergence r is given, one can consider analytic continuations of the series, i.e. analytic functions f which are defined on larger sets than and agree with the given power series on this set. The number r is maximal in the following sense: there always exists a complex number x with x − c = r such that no analytic continuation of the series can be defined at x.
The power series expansion of the inverse function of an analytic function can be determined using the Lagrange inversion theorem.
The sum of a power series with a positive radius of convergence is an analytic function at every point in the interior of the disc of convergence. However, different behavior can occur at points on the boundary of that disc. For example:
1
z=1
z<1
z=1
1
z=1
z=1
1
z=1
1
z=1
See main article: Formal power series. In abstract algebra, one attempts to capture the essence of power series without being restricted to the fields of real and complex numbers, and without the need to talk about convergence. This leads to the concept of formal power series, a concept of great utility in algebraic combinatorics.
An extension of the theory is necessary for the purposes of multivariable calculus. A power series is here defined to be an infinite series of the form
f(x_{1,}...,x_{n)}=
infty  
\sum  
j_{1,}...,j_{n}=0 
a  
j_{1,}...,j_{n} 
n  
\prod  
k=1 
(x_{k}
j_{k}  
c  
k) 
,
where j = (j_{1}, …, j_{n}) is a vector of natural numbers, the coefficients a_{(j1}, …, j_{n}) are usually real or complex numbers, and the center c = (c_{1}, …, c_{n}) and argument x = (x_{1}, …, x_{n}) are usually real or complex vectors. The symbol
\Pi
f(x)=
\sum  
\alpha\inN^{n} 
a_{\alpha}(xc)^{\alpha.}
where
N
N^{n}
The theory of such series is trickier than for singlevariable series, with more complicated regions of convergence. For instance, the power series $\backslash sum\_^\backslash infty\; x\_1^n\; x\_2^n$ is absolutely convergent in the set
\{(x_{1,}x_{2):}x_{1}x_{2}<1\}
(logx_{1,}logx_{2)}
(x_{1,}x_{2)}
Let α be a multiindex for a power series f(x_{1}, x_{2}, …, x_{n}). The order of the power series f is defined to be the least value
r
r=\alpha=\alpha_{1+\alpha}_{2+ … +\alpha}_{n}
infty