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.

References

*cite book |last=Rudin |first=Walter |title=Functional Analysis |year=1991 |month=January |publisher=McGraw-Hill Science/Engineering/Math |id=ISBN 0-07-054236-8
*cite book |last=Rudin |first=Walter |title=Real and Complex Analysis |year=1986 |month=May |publisher=McGraw-Hill Science/Engineering/Math |id=ISBN 0-07-054234-1
*Whittaker and Watson (1927). "A Course in Modern Analysis", fourth edition. Cambridge University Press, p. 49.