Weierstrass M-test

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