Radius of convergence

The radius $R$ of the largest disk around a power series' center on which it converges.

The radius of convergence of a power series $\sum a_n(z-z_0)^n$ is the number $R$ (possibly $0$ or $\infty$ ) such that the series converges absolutely for every $z$ with $|z-z_0|<R$ and diverges for every $z$ with $|z-z_0|>R$ ; behavior exactly on the boundary circle $|z-z_0|=R$ is not determined by $R$ alone and must be checked separately. It is computed from the coefficients via the ratio or root test applied to $|a_n|$ , and it names the open disk on which the power series defines an analytic function.

Examples

$\sum z^n/n!$ has radius of convergence $\infty$ , since the ratio test gives $|a_{n+1}/a_n|=1/(n+1)\to 0$ .
$\sum z^n$ has radius of convergence $1$ : it converges for $|z|<1$ and diverges for $|z|>1$ .