Term-by-term differentiation of power series

Term-by-term differentiation of power series: Let $f(x)=\sum_{n=0}^\infty a_n (x-x_0)^n$ have radius of convergence $R>0$. Define the derived series $\sum_{n=1}^\infty n a_n (x-x_0)^{n-1}.$ Then:

• the derived series has the same radius of convergence $R$, and
• for every $|x-x_0|<R$, the function $f$ is differentiable and $f'(x)=\sum_{n=1}^\infty n a_n (x-x_0)^{n-1}.$

This theorem explains why power series define real-analytic (or complex-analytic) functions on their domain of convergence.

Proof sketch: Fix $r<R$. On $|x-x_0|\le r$, both the original series and the derived series converge uniformly (use M-test ). Use the uniform convergence of the derived series to control difference quotients and justify exchanging limit and summation, yielding the derivative formula for $|x-x_0|<R$.