Weierstrass M-test

Weierstrass M-test: Let $X$ be a set and let $f_n:X\to\mathbb{R}$ (or $\mathbb{C}$). Suppose there exist numbers $M_n\ge 0$ such that $|f_n(x)|\le M_n \quad \text{for all } x\in X \text{ and all } n,$ and the numerical series $\sum_{n=1}^\infty M_n$ converges . Then the function series $\sum_{n=1}^\infty f_n(x)$ converges uniformly on $X$. In particular, it converges absolutely and uniformly: $\sup_{x\in X}\sum_{n=1}^\infty |f_n(x)| < \infty.$

The M-test is a primary mechanism for proving uniform convergence, especially for power series and Fourier-type expansions.

Proof sketch: Use the uniform Cauchy criterion : for $m>n$, $\sup_{x\in X}\left|\sum_{k=n+1}^m f_k(x)\right| \le \sup_{x\in X}\sum_{k=n+1}^m |f_k(x)| \le \sum_{k=n+1}^m M_k.$ Since $\sum M_k$ converges, the tail $\sum_{k=n+1}^m M_k$ can be made arbitrarily small uniformly in $m$, proving uniform convergence.