Weierstrass M-test

Statement

Let $S$ be a set and let $f_n \colon S \to \mathbb{R}$ for $n \ge 0$. Assume that there are real numbers $M_n \ge 0$ with $|f_n(x)| \le M_n$ for all $n$ and all $x \in S$, and $\sum_{n=0}^{\infty} M_n < \infty$. Then the series of functions $\sum_{n=0}^{\infty} f_n$ converges absolutely and uniformly on $S$.

Remarks

Absolute convergence at each $x$ follows from the comparison test. For $m < n$, $|s_n(x) - s_m(x)| \le \sum_{k=m+1}^{n} M_k$. So the partial sums are uniformly Cauchy. For example, $\sum \frac{\sin(n x)}{n^2}$ converges uniformly on $\mathbb{R}$, with $M_n = 1/n^2$. Mathlib states it as tendstoUniformlyOn_tsum.

Explanations

L1 human

If each function is bounded by a number, and these numbers have a finite sum, then the series of functions converges uniformly. The bounds do not depend on $x$, so the convergence is the same everywhere.