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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 of a real variable.
Suppose <math>\{f_n\}<math> is a sequence of complex-valued functions defined on a subset <math>A\subseteq \mathbb{R}<math>, and that for some fixed positive integer N, there exist positive constants <math>M_n<math> such that
- <math>|f_n(x)|\leq M_n<math>
for all <math>n\geq N<math> and all <math>x\in 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>. (See uniform convergence.)
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 afforded by the Banach space.
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