# Weierstrass M-test

Let $X$ be any set, $\{f_{n}\}_{n\in\mathbb{N}}$ a sequence of real or complex valued functions on $X$ and $\{M_{n}\}_{n\in\mathbb{N}}$ a sequence of non-negative real numbers. Suppose that, for each $n\in\mathbb{N}$ and $x\in X$, we have $|f_{n}(x)|\leq M_{n}$. Then $f=\sum_{n=1}^{\infty}f_{n}$ converges uniformly if $\sum_{n=1}^{\infty}M_{n}$ converges.

Title Weierstrass M-test WeierstrassMtest 2013-03-22 12:56:11 2013-03-22 12:56:11 yark (2760) yark (2760) 13 yark (2760) Theorem msc 30A99 AbsoluteConvergence