Dirichlet's test

In mathematics, Dirichlet's test is a method of testing for the convergence of a series. It is named after its author Peter Gustav Lejeune Dirichlet, and was published posthumously in the Journal de Mathématiques Pures et Appliquées in 1862.

Statement

The test states that if is a sequence of real numbers and a sequence of complex numbers satisfying

• • • for every positive integer N

where M is some constant, then the series converges.

Proof

Let and .

From summation by parts, we have that .

Since is bounded by M and , the first of these terms approaches zero, as n.

On the other hand, since the sequence is decreasing, is positive for all k, so . That is, the magnitude of the partial sum of Bn, times a factor, is less than the upper bound of the partial sum Bn (a value M) times that same factor.

But , which is a telescoping series that equals and therefore approaches as n. Thus, converges.

In turn, converges as well by the Direct comparison test. The series converges, as well, by the absolute convergence test. Hence converges.

Applications

A particular case of Dirichlet's test is the more commonly used alternating series test for the case Another corollary is that converges whenever is a decreasing sequence that tends to zero.

Improper integrals

An analogous statement for convergence of improper integrals is proven using integration by parts. If the integral of a function f is uniformly bounded over all intervals, and g is a monotonically decreasing non-negative function, then the integral of fg is a convergent improper integral.

Notes

1. Démonstration d’un théorème d’Abel. Journal de mathématiques pures et appliquées 2nd series, tome 7 (1862), p. 253-255.

References

• Hardy, G. H., A Course of Pure Mathematics, Ninth edition, Cambridge University Press, 1946. (pp. 379–380).
• Voxman, William L., Advanced Calculus: An Introduction to Modern Analysis, Marcel Dekker, Inc., New York, 1981. (§8.B.13-15) ISBN 0-8247-6949-X.