# 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

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

where M is some constant, then the series

$\sum _^{\infty }a_b_$ converges.

## Proof

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

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

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

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

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

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

## Applications

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

$b_=(-1)^\Rightarrow \left|\sum _^b_\right|\leq 1.$ Another corollary is that $\sum _^{\infty }a_\sin n$ converges whenever $\\}$ 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.