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.[1]
Part of a series of articles about  
Calculus  





Specialized 

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 .
On the other hand, since the sequence is decreasing, is nonnegative for all k, so . That is, the magnitude of the partial sum of , times a factor, is less than the upper bound of the partial sum (a value M) times that same factor.
But , which is a telescoping sum that equals and therefore approaches as . 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 nonnegative function, then the integral of fg is a convergent improper integral.
Notes
 Démonstration d’un théorème d’Abel. Journal de mathématiques pures et appliquées 2nd series, tome 7 (1862), p. 253255 Archived 20110721 at the Wayback Machine.
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.1315) ISBN 082476949X.