Weierstrass M-test

Statement

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 non-negative numbers (M**n) satisfying the conditions

• |f_n(x)|\leq M_n for all n \geq 1 and all x \in A, and
• \sum_{n=1}^{\infty} M_n converges. Then the series :\sum_{n=1}^{\infty} f_n (x) converges absolutely and uniformly on A.

A series satisfying the hypothesis is called normally convergent. 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.

Proof

Consider the sequence of functions :S_{n}(x) = \sum_{k=1}^{n}f_{k}(x).

Since the series \sum_{n=1}^{\infty}M_{n} converges and Mn ≥ 0 for every n, then by the Cauchy criterion, :\forall \varepsilon0 : \exists N : \forall mnN : \sum_{k=n+1}^{m}M_{k} For the chosen N, : \forall x \in A : \forall m n N : \left|S_{m}(x)-S_{n}(x)\right|=\left|\sum_{k=n+1}^{m}f_{k}(x)\right|\overset{(1)}{\leq} \sum_{k=n+1}^{m}|f_{k}(x)|\leq \sum_{k=n+1}^{m}M_{k} (Inequality (1) follows from the triangle inequality.)

For each x, the sequence Sn(x) is thus a Cauchy sequence in R or C, and by completeness, it converges to some number S(x) that depends on x. For n  N we can write : \left|S(x) - S_{n}(x)\right|=\left|\lim_{m\to\infty} S_{m}(x) - S_{n}(x)\right|=\lim_{m\to\infty} \left|S_{m}(x) - S_{n}(x)\right|\leq\varepsilon . 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 \sum_{k=1}^{\infty}f_{k}(x) converges uniformly.

Analogously, one can prove that \sum_{k=1}^{\infty}|f_{k}(x)| converges uniformly.

Generalization

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

:|f_n(x)|\leq M_n

is to be replaced by

:|f_n(x)|\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.

References