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.
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Statement
Weierstrass M-test. Suppose that {fn} is a sequence of real- or complex-valued functions defined on a set A, and that there is a sequence of positive numbers {Mn} 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 fn are continuous on A, then the series converges to a continuous function.
Generalization
A more general version of the Weierstrass M-test holds if the codomain of the functions {fn} is any Banach space, in which case the statement
may be replaced by
where
Proof
Consider the sequence of functions
Since the series
For the chosen N,
Thus the sequence of partial sums of the series converges uniformly. Hence, by definition, the series
Inequality (1) follows from the triangle inequality.