# Power Series.

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

Theorem 8.0.1
1. The sum of a power series is an analytic function on the open set .
2. 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.

Example 8.0.2   Let . This is a geometric series whose ratio is equal to . It is convergent for and its sum is equal to .

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

Subsections
Noah Dana-Picard 2007-12-24