# Weierstrass M-test

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

[itex]|f_n(x)|\leq M_n[itex]

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

[itex]\sum_{n=1}^{\infty} M_n[itex]

converges. Then, the series

[itex]\sum_{n=1}^{\infty} f_n (x)[itex]

converges uniformly on [itex]A[itex].

A more general version of the Weierstrass M-test holds if the codomain of the functions [itex]\{f_n\}[itex] is any Banach space, in which case the statement

[itex]|f_n|\leq M_n[itex]

may be replaced by

[itex]||f_n||\leq M_n[itex],

where [itex]||\cdot||[itex] is the norm on the Banach space.

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