Weierstrass M-test
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High Quality Content by WIKIPEDIA articles! 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 {fn} is a sequence of real- or complex-valued functions defined on a set A, and that there exist positive constants Mn such that f_n(x) leq M_n for all n 1 and all x in A. Suppose further that the series sum_{n=1}^{infty} M_n converges. Then, the series sum_{n=1}^{infty} f_n (x) converges uniformly on A. In particular, if the set A is a topological space and...
High Quality Content by WIKIPEDIA articles! 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 {fn} is a sequence of real- or complex-valued functions defined on a set A, and that there exist positive constants Mn such that f_n(x) leq M_n for all n 1 and all x in A. Suppose further that the series sum_{n=1}^{infty} M_n converges. Then, the series sum_{n=1}^{infty} f_n (x) converges uniformly on A. In particular, if the set A is a topological space and the functions fn are continuous on A, the series converges to a continuous function. 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 f_n leq M_n may be replaced by f_n leq M_n, where 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.
