Weierstrass M-test Guide, Meaning , Facts, Information and Description

In mathematics, the Weierstrass M-test is an analogue of the comparison test for infinite series, and applies to a series whose terms are themselves functions of a real variable.

Suppose is a sequence of complex-valued functions defined on a subset , and that for some fixed positive integer N, there exist positive constants such that

for all and all . Suppose further that the series

converges. Then, the series

converges uniformly on . (See uniform convergence.)

A more general version of the Weierstrass M-test holds if the codomain of the functions is any Banach space, in which case the statement

may be replaced by

,

where is the norm afforded by the Banach space.

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