# power series

A power series is a series of the form

 $\sum_{k=0}^{\infty}a_{k}(x-x_{0})^{k},$

with $a_{k},x_{0}\in\mathbb{R}$ or $\in\mathbb{C}$. 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 $\{x\mid|x-x_{0}|, with

 $r=\liminf_{k\to\infty}\frac{1}{\sqrt[k]{|a_{k}|}}$

It is divergent for every $x$ with $|x-x_{0}|>r$. For $|x-x_{0}|=r$ no general predictions can be made. If $r=\infty$, the power series converges absolutely and uniformly for every real or complex $x.$ The real number $r$ is called the of the power series.

Examples of power series are:

• Taylor series, for example:

 $e^{x}=\sum_{k=0}^{\infty}\frac{x^{k}}{k!}.$
• The geometric series:

 $\frac{1}{1-x}=\sum_{k=0}^{\infty}x^{k},$

with $|x|<1$.

Power series have some important :

• If a power series converges for a $z_{0}\in\mathbb{C}$ then it also converges for all $z\in\mathbb{C}$ with $|z-x_{0}|<|z_{0}-x_{0}|$.

• Also, if a power series diverges for some $z_{0}\in\mathbb{C}$ then it diverges for all $z\in\mathbb{C}$ with $|z-x_{0}|>|z_{0}-x_{0}|$.

• For $|x-x_{0}| Power series can be added by adding coefficients and multiplied in the obvious way:

 $\sum_{k=0}^{\infty}a_{k}(x-x_{o})^{k}\cdot\sum_{l=0}^{\infty}b_{j}(x-x_{0})^{j% }=a_{0}b_{0}+(a_{0}b_{1}+a_{1}b_{0})(x-x_{0})+(a_{0}b_{2}+a_{1}b_{1}+a_{2}b_{0% })(x-x_{0})^{2}\ldots.$
• (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 2013-03-22 12:32:55 Last modified on 2013-03-22 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