We consider a power series . Denote by its radius of convergence.

- The sum of a power series is an analytic function on the open set .
- The successive derivatives of are obtained by term-by-term differentiation.

This theorem is a consequence of Corollary 3.6 and Thm 3.7.

Try this with the examples in 1.2.

By term-by-term differentiation, we get the successive series, all convergent for :