Uniform convergence of power series on compact sets

Uniform convergence on compacts: Let $\sum_{n=0}^\infty a_n(x-x_0)^n$ be a power series with radius of convergence $R\in(0,\infty]$. If $K\subset (x_0-R,x_0+R)$ is compact, then the series converges absolutely and uniformly on $K$.

In particular, for every $r$ with $0<r<R$, the series converges absolutely and uniformly on the closed interval $[x_0-r,x_0+r]$.

This is the standard tool behind term-by-term operations on power series, such as term-by-term integration and term-by-term differentiation, and can be proved using the Weierstrass M-test . It is a concrete instance of uniform convergence on compact sets .