Weierstrass M-test

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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 with real or complex values.

Suppose [\] is a sequence of real- or complex-valued functions defined on a set [A], and that there exist positive constants [M_n] such that

[|f_n(x)|\leq M_n]

for all [n]≥[1] and all [x] in [A]. Suppose further that the series

[\sum_^ M_n]

converges. Then, the series

[\sum_^ f_n (x)]

converges uniformly on [A].

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

[|f_n|\leq M_n]

may be replaced by

[||f_n||\leq M_n],

where [||\cdot||] is the norm on the Banach space. For an example of the use of this test on a Banach space, see the article Fréchet derivative.

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