radius of convergence
To the power series
(1) |
there exists a number , its radius of convergence, such that the series converges absolutely for all (real or complex) numbers with and diverges whenever . This is known as Abel’s theorem on power series. (For no general statements can be made.)
The radius of convergence is given by:
(2) |
and can also be computed as
(3) |
if this limit exists.
It follows from the Weierstrass -test (http://planetmath.org/WeierstrassMTest) that for any radius smaller than the radius of convergence, the power series converges uniformly within the closed disk of radius .
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 |