 Dirichlet's Test
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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 $\{a_{n}\}$ is a sequence of real numbers and $\{b_{n}\}$ a sequence of complex numbers satisfying

• $\{a_{n}\}$ is monotonic
• $\lim _{n\rightarrow \infty }a_{n}=0$ • $\left|\sum _{n=1}^{N}b_{n}\right|\leq M$ for every positive integer N

where M is some constant, then the series

$\sum _{n=1}^{\infty }a_{n}b_{n}$ converges.

## Proof

Let $S_{n}=\sum _{k=1}^{n}a_{k}b_{k}$ and $B_{n}=\sum _{k=1}^{n}b_{k}$ .

From summation by parts, we have that $S_{n}=a_{n}B_{n}+\sum _{k=1}^{n-1}B_{k}(a_{k}-a_{k+1})$ . Since $B_{n}$ is bounded by M and $a_{n}\rightarrow 0$ , the first of these terms approaches zero, $a_{n}B_{n}\to 0$ as $n\to \infty$ .

We have, for each k, $|B_{k}(a_{k}-a_{k+1})|\leq M|a_{k}-a_{k+1}|$ . But, if $\{a_{n}\}$ is decreasing,

$\sum _{k=1}^{n}M|a_{k}-a_{k+1}|=\sum _{k=1}^{n}M(a_{k}-a_{k+1})=M\sum _{k=1}^{n}(a_{k}-a_{k+1})$ ,

which is a telescoping sum, that equals $M(a_{1}-a_{n+1})$ and therefore approaches $Ma_{1}$ as $n\to \infty$ . Thus, $\sum _{k=1}^{\infty }M(a_{k}-a_{k+1})$ converges. And, if $\{a_{n}\}$ is increasing,

$\sum _{k=1}^{n}M|a_{k}-a_{k+1}|=-\sum _{k=1}^{n}M(a_{k}-a_{k+1})=-M\sum _{k=1}^{n}(a_{k}-a_{k+1})$ ,

which is again a telescoping sum, that equals $-M(a_{1}-a_{n+1})$ and therefore approaches $-Ma_{1}$ as $n\to \infty$ . Thus, again, $\sum _{k=1}^{\infty }M(a_{k}-a_{k+1})$ converges.

So, $\sum _{k=1}^{\infty }|B_{k}(a_{k}-a_{k+1})|$ converges as well by the direct comparison test. The series $\sum _{k=1}^{\infty }B_{k}(a_{k}-a_{k+1})$ converges, as well, by the absolute convergence test. Hence $S_{n}$ converges.

## Applications

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

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