Weierstrass M-test

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

${\displaystyle \forall n\geq 1,\forall x\in A:\ |f_{n}(x)|\leq M_{n},}$

${\displaystyle \sum _{n=1}^{\infty }M_{n}<\infty .}$

Then the series

${\displaystyle \sum _{n=1}^{\infty }f_{n}(x)}$

converges absolutely and uniformly on A.

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.

Consider the sequence of functions

${\displaystyle S_{n}(x)=\sum _{k=1}^{n}f_{k}(x).}$

Since the series ${\displaystyle \sum _{n=1}^{\infty }M_{n}}$ converges and M_n ≥ 0 for every n, then by the Cauchy criterion,

${\displaystyle \forall \varepsilon >0:\exists N:\forall m>n>N:\sum _{k=n+1}^{m}M_{k}<\varepsilon .}$

For the chosen N,

${\displaystyle \forall x\in A:\forall m>n>N}$

${\displaystyle \left|S_{m}(x)-S_{n}(x)\right|=\left|\sum _{k=n+1}^{m}f_{k}(x)\right|{\overset {(1)}{\leq }}\sum _{k=n+1}^{m}|f_{k}(x)|\leq \sum _{k=n+1}^{m}M_{k}<\varepsilon .}$

(Inequality (1) follows from the triangle inequality.)

The sequence S_n(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

${\displaystyle \left|S(x)-S_{n}(x)\right|=\left|\lim _{m\to \infty }S_{m}(x)-S_{n}(x)\right|=\lim _{m\to \infty }\left|S_{m}(x)-S_{n}(x)\right|\leq \varepsilon .}$

Since N does not depend on x, this means that the sequence S_n of partial sums converges uniformly to the function S. Hence, by definition, the series ${\displaystyle \sum _{k=1}^{\infty }f_{k}(x)}$ converges uniformly.

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

A more general version of the Weierstrass M-test holds if the common codomain of the functions (f_n) is a Banach space, in which case the premise

${\displaystyle |f_{n}(x)|\leq M_{n}}$

is to be replaced by

${\displaystyle \|f_{n}(x)\|\leq M_{n}}$,

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.

• Example of Weierstrass M-test

• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.
• Rudin, Walter (May 1986). Real and Complex Analysis. McGraw-Hill Science/Engineering/Math. ISBN 0-07-054234-1.
• Rudin, Walter (1976). Principles of Mathematical Analysis. McGraw-Hill Science/Engineering/Math.
• Whittaker, E.T.; Watson, G.N. (1927). A Course in Modern Analysis (Fourth ed.). Cambridge University Press. p. 49.