Weierstrass M-test

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).

StatementEdit

Weierstrass M-test. Suppose that (fn) is a sequence of real- or complex-valued functions defined on a set A, and that there is a sequence of non-negative numbers (Mn) satisfying the conditions

  •   for all   and all  , and
  •   converges.

Then the series

 

converges absolutely and uniformly on A.

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 fn are continuous on A, then the series converges to a continuous function.

ProofEdit

Consider the sequence of functions

 

Since the series   converges and Mn ≥ 0 for every n, then by the Cauchy criterion,

 

For the chosen N,

 
 

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

The sequence Sn(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

 

Since N does not depend on x, this means that the sequence Sn of partial sums converges uniformly to the function S. Hence, by definition, the series   converges uniformly.

Analogously, one can prove that   converges uniformly.

GeneralizationEdit

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

 

is to be replaced by

 ,

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 alsoEdit

ReferencesEdit

  • Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 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.