In mathematics, the Weierstrass M-test is a test for determining whether an infinite series of functions converges uniformly and absolutely.
It applies to series whose terms are bounded functions with real or complex values, and is analogous to the comparison test for determining the convergence of series of real or complex numbers.
Suppose that (fn) is a sequence of real- or complex-valued functions defined on a set A, and that there is a sequence of non-negative numbers (Mn) satisfying the conditions Then the series converges absolutely and uniformly on A.
A series satisfying the hypothesis is called normally convergent.
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.
For n > N we can write Since N does not depend on x, this means that the sequence Sn of partial sums converges uniformly to the function S. Hence, by definition, the series
A more general version of the Weierstrass M-test holds if the common codomain of the functions (fn) is a Banach space, in which case the premise is to be replaced by where
For an example of the use of this test on a Banach space, see the article Fréchet derivative.