# 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]

## Statement

The test states that if ${\displaystyle \\}}$ is a sequence of real numbers and ${\displaystyle \\}}$ a sequence of complex numbers satisfying

• ${\displaystyle a_\leq a_}$
• ${\displaystyle \lim _a_=0}$
• ${\displaystyle \left|\sum _^b_\right|\leq M}$ for every positive integer N

where M is some constant, then the series

${\displaystyle \sum _^{\infty }a_b_}$

converges.

## Proof

Let ${\displaystyle S_=\sum _^a_b_}$ and ${\displaystyle B_=\sum _^b_}$.

From summation by parts, we have that ${\displaystyle S_=a_B_+\sum _^B_(a_-a_)}$.

Since ${\displaystyle B_}$ is bounded by M and ${\displaystyle a_\rightarrow 0}$, the first of these terms approaches zero, ${\displaystyle a_B_\to 0}$ as ${\displaystyle n\to \infty }$.

On the other hand, since the sequence ${\displaystyle a_}$ is decreasing, ${\displaystyle a_-a_}$ is non-negative for all k, so ${\displaystyle |B_(a_-a_)|\leq M(a_-a_)}$. That is, the magnitude of the partial sum of ${\displaystyle B_}$, times a factor, is less than the upper bound of the partial sum ${\displaystyle B_}$ (a value M) times that same factor.

But ${\displaystyle \sum _^M(a_-a_)=M\sum _^(a_-a_)}$, which is a telescoping sum that equals ${\displaystyle M(a_-a_)}$ and therefore approaches ${\displaystyle Ma_}$ as ${\displaystyle n\to \infty }$. Thus, ${\displaystyle \sum _^{\infty }M(a_-a_)}$ converges.

In turn, ${\displaystyle \sum _^{\infty }|B_(a_-a_)|}$ converges as well by the direct comparison test. The series ${\displaystyle \sum _^{\infty }B_(a_-a_)}$ converges, as well, by the absolute convergence test. Hence ${\displaystyle S_}$ converges.

## Applications

A particular case of Dirichlet's test is the more commonly used alternating series test for the case

${\displaystyle b_=(-1)^\Rightarrow \left|\sum _^b_\right|\leq 1.}$

Another corollary is that ${\displaystyle \sum _^{\infty }a_\sin n}$ converges whenever ${\displaystyle \\}}$ is a decreasing sequence that tends to zero.

Moreover, Abel's test can be considered a special case of Dirichlet's test.

## 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 Archived 2011-07-21 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.13-15) ISBN 0-8247-6949-X.