Let $f$ be an analytic function defined in a disk of radius $R$ about a point $z_0 \in \mathbb{C}$ . Then the radius of convergence of the Taylor series of $f$ about $z_0$ is at least $R$ .

For example, the function $a(z) = 1 / (1 - z)^2$ is analytic inside the disk $|z| < 1$ . Hence its the radius of covergence of its Taylor series about $0$ is at least $1$ . By direct examination of the Taylor series we can see that its radius of convergence is, in fact, equal to $1$ .

Colloquially, this theorem is stated in the sometimes imprecise but memorable form ``The radius of convergence of the Taylor series is the distance to the nearest singularity.''