Suppose has convergence radius 1. If all , then is a singular point of .

Expand as Taylor series at

If has holomorphic extension on , then the above series must converge for some , , small enough,

Since the double series has positive terms, we can interchange the order of two summations without loss of convergence. Hence the series,

must converge. This contradicts the fact that has convergence radius 1.

- This theorem is posted by Alfred Pringsheim. But his proof is not right. For correct proof, refer to
*Analytic function Volume 1*by Einar Hille. - This theorem concludes there is at least one singular point fo the power series on its circle of convergence.
- Consider , is real, which has convergence radius 1. This theorem applies and shows that is a singular point of . actually, it is the only singularity on . In this case, , as when but not for . as when but not for .