Let be an analytic function defined in a disk of radius about a point . Then the radius of convergence of the Taylor series of about is at least .

For example, the function is analytic inside the disk . Hence its the radius of covergence of its Taylor series about 0 is at least . By direct examination of the Taylor series we can see that its radius of convergence is, in fact, equal to .

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.”