Term-by-term differentiation for power series

Term-by-term differentiation for power series: Let $\sum_{n=0}^\infty a_n (x-x_0)^n$ be a power series with radius of convergence $R>0$ , and define

f(x)=\sum_{n=0}^\infty a_n (x-x_0)^n \quad\text{for }|x-x_0|<R.

Then $f$ is differentiable for $|x-x_0|<R$ , and for every $|x-x_0|<R$ ,

f'(x)=\sum_{n=1}^\infty n\,a_n (x-x_0)^{n-1}.

The differentiated power series has the same radius of convergence $R$ .

Remarks

This theorem underlies the fact that power series define very smooth functions (see power series is analytic on its disk of convergence) and is part of the broader toolkit of term-by-term operations for power series.