radius of convergence of a complex function

Let $f$ be an analytic function defined in a disk of radius $R$ about a point $z_0\in ℂ$. 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 . 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.”