To the power series \begin{equation} \sum_{k=0}^{\infty}a_k(x-x_0)^k \end{equation}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 $|x-x_0|<r$ 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: \begin{equation} r=\liminf_{k\to\infty}\frac{1}{\sqrt[k]{|a_k|}} \end{equation}and can also be computed as \begin{equation} r=\lim_{k\to\infty}\left|\frac{a_k}{a_{k+1}}\right|, \end{equation}if this limit exists.

It follows from the Weierstrass $M$ test that for any radius $r'$ smaller than the radius of convergence, the power series converges uniformly within the closed disk of radius $r'$