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

<math>|f_n(x)|\leq M_n<math>

for all <math>n<math>≥<math>1<math> and all <math>x<math> in <math>A<math>. Suppose further that the series

<math>\sum_{n=1}^{\infty} M_n<math>

converges. Then, the series

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

converges uniformly on <math>A<math>.

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

<math>|f_n|\leq M_n<math>

may be replaced by

<math>||f_n||\leq M_n<math>,

where <math>||\cdot||<math> is the norm on the Banach space.