# Weierstrass’ criterion of uniform convergence

###### Theorem.

Let the real functions $f_{1}(x)$, $f_{2}(x)$, … be defined in the interval $[a,b]$.  If they all the condition

 $|f_{n}(x)|\leqq M_{n}\quad\forall\,x\in[a,b],$

with $\sum_{n=1}^{\infty}M_{n}$ a convergent series of , then the function series

 $f_{1}(x)\!+\!f_{2}(x)\!+\!\cdots$

converges uniformly (http://planetmath.org/SumFunctionOfSeries) on the interval $[a,b]$.

The theorem is valid also for the series with complex function terms, when one replaces the interval with a subset of $\mathbb{C}$.

Title Weierstrass’ criterion of uniform convergence WeierstrassCriterionOfUniformConvergence 2013-03-22 14:38:21 2013-03-22 14:38:21 pahio (2872) pahio (2872) 9 pahio (2872) Theorem msc 26A15 msc 40A30 Weierstrass’ M-test