In mathematics, the Weierstrass M-test is a test for showing that an infinite series of functions converges uniformly. It applies to series whose terms are functions with real or complex values, and is analogous to the comparison test for determining the convergence of series of real or complex numbers.
Then the series
converges uniformly on A.
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.
may 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.
Let M be the limit of the sum . Since the sum is absolutely convergent, call its limit f(x).
By convergence of the M sum, for ε > 0 there exists an integer K such that
We will show that converges uniformly by showing that
The crucial point here is that K does not depend on x.
See also 
- 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.
- Rudin, Walter (1976). Principles of Mathematical Analysis. McGraw-Hill Science/Engineering/Math.
- Whittaker and Watson (1927). A Course in Modern Analysis, fourth edition. Cambridge University Press, p. 49.