PlanetMath: power series

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power series

(Definition)

A power series is a series of the form

$\displaystyle \sum_{k=0}^{\infty}a_k(x-x_0)^k,$

with $a_k,x_0\in\mathbb{R}$ or $\in\mathbb{C}$ . The

are called the coefficients and

the center of the power series.

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 where it converges to . In addition it is absolutely and uniformly convergent in the region $\{x\mid \vert x-x_0\vert<r\}$ , with

$\displaystyle r=\liminf_{k\to\infty}\frac{1}{\sqrt[k]{\vert a_k\vert}}$

It is divergent for every

with $\vert x-x_0\vert > r$ . For $\vert x-x_0\vert= r$ no general predictions can be made. If $r=\infty$ , the power series converges absolutely and uniformly for every real or complex

The real number

is called the radius of convergence of the power series.

Examples of power series are:

Taylor series, for example:
$\displaystyle e^x=\sum_{k=0}^{\infty}\frac{x^k}{k!}.$
The geometric series:
$\displaystyle \frac{1}{1-x}=\sum_{k=0}^{\infty}x^k,$
with $\vert x\vert<1$ .

Power series have some important properties:

If a power series converges for a $z_0\in\mathbb{C}$ then it also converges for all $z\in\mathbb{C}$ with $\vert z-x_0\vert<\vert z_0-x_0\vert$ .
Also, if a power series diverges for some $z_0\in\mathbb{C}$ then it diverges for all $z\in\mathbb{C}$ with $\vert z-x_0\vert>\vert z_0-x_0\vert$ .
For $\vert x-x_0\vert<r$ Power series can be added by adding coefficients and multiplied in the obvious way:
$\displaystyle \sum_{k=0}^\infty a_k(x-x_o)^k\cdot\sum_{l=0}^\infty b_j(x-x_0)^j = a_0b_0+(a_0b_1+a_1b_0)(x-x_0)+(a_0b_2+a_1b_1+a_2b_0)(x-x_0)^2\ldots.$
(Uniqueness) If two power series are equal and their centers are the same, then their coefficients must be equal.
Power series can be termwise differentiated and integrated. These operations keep the radius of convergence.

"power series" is owned by azdbacks4234. [ full author list (3) | owner history (3) ]

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Also defines:

constant term

Attachments:: series inversion (Derivation) by stevecheng
identity theorem of power series (Theorem) by pahio

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Cross-references: operations, obvious, diverges, geometric series, Taylor series, radius of convergence, complex, real, converges absolutely, divergent, region, uniformly convergent, addition, convergent, function, converges, coefficients, series
There are 75 references to this entry.

This is version 20 of power series, born on 2002-03-19, modified 2008-03-26.
Object id is 2793, canonical name is PowerSeries.
Accessed 16070 times total.

Classification:

AMS MSC:	30B10 (Functions of a complex variable :: Series expansions :: Power series )
	40A30 (Sequences, series, summability :: Convergence and divergence of infinite limiting processes :: Convergence and divergence of series and sequences of functions)

Pending Errata and Addenda

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