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 is a sequence of real- or complex-valued functions defined on a set , and that there exist positive constants such that
:
for all ≥ and all in . Suppose further that the series
:
converges. Then, the series
:
converges uniformly on . 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
:
may be replaced by
:,
where is the norm on the Banach space.
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