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 $|f_n(x)|\le M_n$. Then $f={\sum }_{n=1}^{\mathrm{\infty }}{f}_n$ converges uniformly if ${\sum }_{n=1}^{\mathrm{\infty }}{M}_n$ converges.

Title	Weierstrass M-test
Canonical name	WeierstrassMtest
Date of creation	2013-03-22 12:56:11
Last modified on	2013-03-22 12:56:11
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	13
Author	yark (2760)
Entry type	Theorem
Classification	msc 30A99
Related topic	AbsoluteConvergence