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

• is monotonic
• 
• 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 .

We have, for each k, .

Since is monotone, it is either decreasing or increasing:

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

So, the series converges, 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. To see that is bounded, we can use the summation formula

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 non-negative monotonically decreasing function, then the integral of fg is a convergent improper integral.