power series

A power series is a series of the form

$$\sum _{k=0}^{\mathrm{\infty }}{a}_k{(x-x_0)}^k,$$

with $a_k,x_0\in ℝ$ or $\in ℂ$. 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 , 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:

  $$e^x=\sum _{k=0}^{\mathrm{\infty }}\frac{x^k}{k!}.$$

• The geometric series:

  $$\frac{1}{1-x}=\sum _{k=0}^{\mathrm{\infty }}{x}^k,$$

  with .

Power series have some important :

• If a power series converges for a $z_0\in ℂ$ then it also converges for all $z\in ℂ$ with .

• Also, if a power series diverges for some $z_0\in ℂ$ then it diverges for all $z\in ℂ$ 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	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