Uniform convergence of power series on compact subsets

Uniform convergence of power series on compact subsets: Let $\sum_{n=0}^\infty a_n (x-x_0)^n$ be a power series with radius of convergence $R>0$. Then for every $r$ with $0<r<R$, the series converges uniformly on the closed set $\{x:|x-x_0|\le r\}.$

Uniform convergence on compact subsets is the mechanism behind the “good behavior” of power series: continuity , term-by-term differentiation , and term-by-term integration hold inside the disk/interval of convergence.

Proof sketch: For $|x-x_0|\le r$, $|a_n(x-x_0)^n|\le |a_n|\,r^n.$ Since $r<R$, the numerical series $\sum |a_n|r^n$ converges (absolute convergence inside the radius). Apply the Weierstrass M-test to obtain uniform convergence.