Weierstrass M-test

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 ${f n}$ is a sequence of real- or complex-valued functions defined on a set $A$ , and that there exist positive constants $M n$ 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 $A$ .

A more general version of the Weierstrass M-test holds if the codomain of the functions ${f n}$ 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.

[edit] References

Rudin, Walter (January 1991). Functional Analysis. McGraw-Hill Science/Engineering/Math. ISBN 0-07-054236-8.
Rudin, Walter (May 1986). Real and Complex Analysis. McGraw-Hill Science/Engineering/Math. ISBN 0-07-054234-1.
Whittaker and Watson (1927). A Course in Modern Analysis, fourth edition. Cambridge University Press, p. 49.