Term-by-term differentiation for power series

Inside the radius of convergence, a power series can be differentiated by differentiating each term.

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 $R$ .

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.