Power series is analytic on its disk of convergence

Power series is analytic on its disk of convergence: Let $\sum_{n=0}^\infty a_n (z-z_0)^n$ be a power series with radius of convergence $R>0$ . Define

f(z)=\sum_{n=0}^\infty a_n (z-z_0)^n \quad\text{for }|z-z_0|<R.

Then $f$ is complex differentiable on the open disk $\{z:|z-z_0|<R\}$ , and for every $|z-z_0|<R$ ,

f'(z)=\sum_{n=1}^\infty n\,a_n (z-z_0)^{n-1}.

Remarks

This is the fundamental analytic regularity of power series and is typically established via term-by-term differentiation together with uniform convergence on compact subsets (see uniform convergence on compacts for power series).