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).
Then the series
Remark. 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.
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.)
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.
is to be replaced by
- Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. 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.