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Power series is analytic on its disk of convergence

Within its radius of convergence, a power series defines a function with derivatives given by termwise differentiation.

Power series is analytic on its disk of convergence: Let ∑n=0∞an(z−z0)n\sum_{n=0}^\infty a_n (z-z_0)^n be a with radius of convergence R>0R>0. Define

f(z)=∑n=0∞an(z−z0)nfor ∣z−z0∣<R.f(z)=\sum_{n=0}^\infty a_n (z-z_0)^n \quad\text{for }|z-z_0|<R.

Then ff is complex differentiable on the open disk {z:∣z−z0∣<R}\{z:|z-z_0|<R\}, and for every ∣z−z0∣<R|z-z_0|<R,

f′(z)=∑n=1∞n an(z−z0)n−1.f'(z)=\sum_{n=1}^\infty n\,a_n (z-z_0)^{n-1}.
Remarks

This is the fundamental analytic regularity of and is typically established via together with uniform convergence on compact subsets (see ).