power series
A power series^{} is a series of the form
$$\sum _{k=0}^{\mathrm{\infty}}{a}_{k}{(x{x}_{0})}^{k},$$ 
with ${a}_{k},{x}_{0}\in \mathbb{R}$ or $\in \u2102$. The ${a}_{k}$ are called the coefficients and ${x}_{0}$ the center of the power series. ${a}_{0}$ is called the constant term.
Where it converges^{} the power series defines a function, which can thus be represented by a power series. This is what power series are usually used for. Every power series is convergent at least at $x={x}_{0}$ where it converges to ${a}_{0}$. In addition it is absolutely and uniformly convergent in the region $$, with
$$r=\underset{k\to \mathrm{\infty}}{lim\; inf}\frac{1}{\sqrt[k]{{a}_{k}}}$$ 
It is divergent for every $x$ with $x{x}_{0}>r$. For $x{x}_{0}=r$ no general predictions can be made. If $r=\mathrm{\infty}$, the power series converges absolutely and uniformly for every real or complex $x.$ The real number $r$ is called the radius of convergence^{} of the power series.
Examples of power series are:

•
Taylor series^{}, for example:
$${e}^{x}=\sum _{k=0}^{\mathrm{\infty}}\frac{{x}^{k}}{k!}.$$  •
Power series have some important :

•
If a power series converges for a ${z}_{0}\in \u2102$ then it also converges for all $z\in \u2102$ with $$.

•
Also, if a power series diverges for some ${z}_{0}\in \u2102$ then it diverges for all $z\in \u2102$ with $z{x}_{0}>{z}_{0}{x}_{0}$.

•
For $$ Power series can be added by adding coefficients and multiplied in the obvious way:
$$\sum _{k=0}^{\mathrm{\infty}}{a}_{k}{(x{x}_{o})}^{k}\cdot \sum _{l=0}^{\mathrm{\infty}}{b}_{j}{(x{x}_{0})}^{j}={a}_{0}{b}_{0}+({a}_{0}{b}_{1}+{a}_{1}{b}_{0})(x{x}_{0})+({a}_{0}{b}_{2}+{a}_{1}{b}_{1}+{a}_{2}{b}_{0}){(x{x}_{0})}^{2}\mathrm{\dots}.$$ 
•
(Uniqueness) If two power series are equal and their are the same, then their coefficients must be equal.

•
Power series can be termwise differentiated and integrated. These operations^{} keep the radius of convergence.
Title  power series 
Canonical name  PowerSeries 
Date of creation  20130322 12:32:55 
Last modified on  20130322 12:32:55 
Owner  azdbacks4234 (14155) 
Last modified by  azdbacks4234 (14155) 
Numerical id  23 
Author  azdbacks4234 (14155) 
Entry type  Definition 
Classification  msc 40A30 
Classification  msc 30B10 
Related topic  TaylorSeries 
Related topic  FormalPowerSeries 
Related topic  TermwiseDifferentiation 
Related topic  AbelsLimitTheorem 
Defines  constant term 