Weierstrass M-test

The Weierstrass M-test is a useful result in studying convergence of function series.

In mathematics, the Weierstrass M-test is a test for showing that an infinite series of functions converges uniformly.

The Weierstrass M-test requires us to find an upper bound on the terms of the series, with independent of the position in the disc:

The Weierstrass M-test says the series converges uniformly, and thus the interchange of the sum and the integral is justified.

The series expansion of the exponential function can be shown to be uniformly convergent on any bounded subset S of using the Weierstrass M-test.

Since for any each term of the series is less than or equal to a in absolute value, the series uniformly converges to a continuous, strictly increasing function g(x), due to the Weierstrass M-test.

In mathematics, the 'Weierstrass M-test' is an analogue of the comparison test for infinite series, and applies to a Series (mathematics) whose terms are themselves function (mathematics) with real number or complex number values.