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

\{f_n\}

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

A

. Suppose further that the series

\sum_{n=1}^{\infty} M_n

converges. Then, the series

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

converges uniformly on

A

. A more general version of the Weierstrass M-test holds if the codomain of the functions

\{f_n\}

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.

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