Weierstrass M-test - Knowlpedia

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

|f_n(x)|\le M_n \quad \text{for all }x\in E\text{ and all }n,

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.

Remarks

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.