To the power series

 $\sum_{k=0}^{\infty}a_{k}(x-x_{0})^{k}$ (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 $x$ with $|x-x_{0}| and diverges whenever $|x-x_{0}|>r$. This is known as Abel’s theorem on power series. (For $|x-x_{0}|=r$ no general statements can be made.)

The radius of convergence is given by:

 $r=\liminf_{k\to\infty}\frac{1}{\sqrt[k]{|a_{k}|}}$ (2)

and can also be computed as

 $r=\lim_{k\to\infty}\left|\frac{a_{k}}{a_{k+1}}\right|,$ (3)

if this limit exists.

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

Title radius of convergence RadiusOfConvergence 2013-03-22 12:32:59 2013-03-22 12:32:59 PrimeFan (13766) PrimeFan (13766) 13 PrimeFan (13766) Theorem msc 40A30 msc 30B10 Abel’s theorem on power series ExampleOfAnalyticContinuation NielsHenrikAbel