PlanetMath: radius of convergence

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radius of convergence

(Theorem)

To the power series

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

(1)

there exists a number $r\in [0,\infty]$ , its radius of convergence, such that the series converges absolutely for all (real or complex) numbers

with $\vert x-x_0\vert<r$ and diverges whenever $\vert x-x_0\vert>r$ . This is known as Abel's theorem on power series. (For $\vert x-x_0\vert= r$ no general statements can be made, except that there always exists at least one complex number

with $\vert x-x_0\vert=r$ such that the series diverges.)

The radius of convergence is given by:

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

(2)

and can also be computed as

$\displaystyle r=\lim_{k\to\infty}\left\vert\frac{a_k}{a_{k+1}}\right\vert,$

(3)

if this limit exists.

It follows from the Weierstrass -test that for any radius smaller than the radius of convergence, the power series converges uniformly within the closed disk of radius .

"radius of convergence" is owned by PrimeFan. [ full author list (3) | owner history (3) ]

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Other names:

Abel's theorem on power series

Attachments:: proof of radius of convergence (Proof) by mathwizard

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Cross-references: closed, converges uniformly, radius, limit, complex number, diverges, complex, real, converges absolutely, series, power series
There are 19 references to this entry.

This is version 9 of radius of convergence, born on 2002-03-19, modified 2008-05-15.
Object id is 2794, canonical name is RadiusOfConvergence.
Accessed 10402 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

1. radius of convergence by ogu on 2008-05-09 14:29:54
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