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]


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

  • is monotonic
  • for every positive integer N

where M is some constant, then the series



Let and .

From summation by parts, we have that . Since is bounded by M and , the first of these terms approaches zero, as .

We have, for each k, . But, if is decreasing,


which is a telescoping sum, that equals and therefore approaches as . Thus, converges. And, if is increasing,


which is again a telescoping sum, that equals and therefore approaches as . Thus, again, converges.

So, converges as well by the direct comparison test. The series converges, as well, by the absolute convergence test. Hence converges.


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.


  1. ^ Démonstration d’un théorème d’Abel. Journal de mathématiques pures et appliquées 2nd series, tome 7 (1862), pp. 253–255 Archived 2011-07-21 at the Wayback Machine.


  • 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.

External links

  • Proof at PlanetMath.org