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.
. 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