Weierstrass M-test explained

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. It is named after the German mathematician Karl Weierstrass (1815-1897).

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 ^{inf ty} _{n =1} M _{n} converges.Then the series \sum ^{inf ty} _{n =1} 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 f_n are continuous on A, then the series converges to a continuous function.

Proof

Consider the sequence of functions

S _{n} (x) = \sum ^{n} _{k =1} f _{k} (x).

Since the series

\sum ^{inf ty} _{n =1} M _{n} converges and for every, then by the Cauchy criterion,\forall \varepsilon> 0 : \exists N : \forall m>n>N : \sum ^{m} _{k = n +1} M _{k} <\varepsilon. For the chosen,\forall x \in A : \forall m> n> N \left|S _{m} (x) - S _{n} (x)\right| = \left|\sum ^{m} _{k = n +1} f _{k} (x)\right|\overset{( 1 )}{\leq} \sum ^{m} _{k = n +1} |f _{k} (x)|\leq \sum ^{m} _{k = n +1} M _{k} <\varepsilon . (Inequality (1) follows from the triangle inequality.)

The sequence is thus a Cauchy sequence in R or C, and by completeness, it converges to some number that depends on x. For n&thinsp;>&thinsp;N we can write

\left|S(x) - S _{n} (x)\right| = \left|\lim _{m\toinfty} S _{m} (x) - S _{n} (x)\right| = \lim _{m\toinfty} \left|S _{m} (x) - S _{n} (x)\right|\leq\varepsilon . Since N does not depend on x, this means that the sequence of partial sums converges uniformly to the function S. Hence, by definition, the series \sum ^{inf ty} _{k =1} f _{k} (x) converges uniformly.

Analogously, one can prove that

\sum ^{inf ty} _{k =1} |f _{k} (x)| converges uniformly.

Generalization

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

|f _{n(x)|\leq} M _{n}

is to be replaced by

\|f _{n(x)\|\leq} M _{n} ,

where

\| ⋅ \| 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.

See also

• Example of Weierstrass M-test

References

• • Book: Rudin, Walter . Real and Complex Analysis . May 1986 . McGraw-Hill Science/Engineering/Math . 0-07-054234-1.
• Book: Rudin, Walter . Principles of Mathematical Analysis . registration . 1976 . McGraw-Hill Science/Engineering/Math.
• Book: Whittaker, E.T. . E. T. Whittaker . Watson . G.N. . G. N. Watson . 1927 . A Course in Modern Analysis . Fourth . Cambridge University Press . 49 .