radius of convergence

To the power series

$$\sum _{k=0}^{\mathrm{\infty }}{a}_k{(x-x_0)}^k$$(1)

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

$$r=\underset{k\to \mathrm{\infty }}{lim\; inf}\frac{1}{\sqrt[k]{|a_k|}}$$(2)

and can also be computed as

$$r=\underset{k\to \mathrm{\infty }}{lim}\left|\frac{a_k}{a_{k+1}}\right|,$$(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 $r^{\prime }$.