Weierstrass M-test

Weierstrass M-test: Let $E$ be a set and let $f_n:E\to\mathbb{R}$ (or $\mathbb{C}$) be functions. Assume there exist numbers $M_n\ge 0$ such that

and the numerical series $\sum_{n=1}^\infty M_n$ is a convergent series . Then the series of functions $\sum_{n=1}^\infty f_n$ converges uniformly on $E$. Moreover, for each $x\in E$ the series $\sum_{n=1}^\infty |f_n(x)|$ converges.

This is a standard sufficient condition for uniform convergence, phrased in terms of bounding the tails of the partial sums by an ordinary numerical series.