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# 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 satisfy 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 constant terms, then the function series

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

converges uniformly 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 $\mathbb{C}$.

Synonym:

Weierstrass' M-test

Type of Math Object:

Theorem

Major Section:

Reference

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## Mathematics Subject Classification

26A15*no label found*40A30

*no label found*

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f_i's undefined by matte ✓