Weierstrass M-test

In mathematics, the Weierstrass M-test is a test for testing whether an infinite series of functions converges uniformly and absolutely. It applies to series whose terms are functions with real or complex values, and is analogous to the comparison test for determining the convergence of series of real or complex numbers.

Weierstrass M-test. Suppose that {f_n} is a sequence of real- or complex-valued functions defined on a set A, and that there is a sequence of positive numbers {M_n} satisfying

Then the series

Remark. The result is often used in combination with the uniform limit theorem. Together they say that if, in addition to the above conditions, the set A is a topological space and the functions f_n are continuous on A, then the series converges to a continuous function.

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

may be replaced by

where ${\displaystyle \|\cdot \|}$ is the norm on the Banach space. For an example of the use of this test on a Banach space, see the article Fréchet derivative.

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