radius of convergence

To the power seriesMathworldPlanetmath

k=0ak(x-x0)k (1)

there exists a number r[0,], its radius of convergenceMathworldPlanetmath, such that the series converges absolutely for all (real or complex) numbers x with |x-x0|<r and diverges whenever |x-x0|>r. This is known as Abel’s theorem on power series. (For |x-x0|=r no general statements can be made.)

The radius of convergence is given by:

r=lim infk1|ak|k (2)

and can also be computed as

r=limk|akak+1|, (3)

if this limit exists.

It follows from the Weierstrass M-test (http://planetmath.org/WeierstrassMTest) that for any radius r smaller than the radius of convergence, the power series converges uniformly within the closed disk of radius r.

Title radius of convergence
Canonical name RadiusOfConvergence
Date of creation 2013-03-22 12:32:59
Last modified on 2013-03-22 12:32:59
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 13
Author PrimeFan (13766)
Entry type Theorem
Classification msc 40A30
Classification msc 30B10
Synonym Abel’s theorem on power series
Related topic ExampleOfAnalyticContinuation
Related topic NielsHenrikAbel