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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.[1]

## Statement

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

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

where M is some constant, then the series

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

converges.

## Proof

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

From summation by parts, we have that ${\displaystyle S_{n}=a_{n+1}B_{n}+\sum _{k=1}^{n}B_{k}(a_{k}-a_{k+1})}$.

Since ${\displaystyle B_{n}}$ is bounded by M and ${\displaystyle a_{n}\rightarrow 0}$, the first of these terms approaches zero, ${\displaystyle a_{n+1}B_{n}\to 0}$ as n→∞.

On the other hand, since the sequence ${\displaystyle a_{n}}$ is decreasing, ${\displaystyle a_{k}-a_{k+1}}$ is positive for all k, so ${\displaystyle |B_{k}(a_{k}-a_{k+1})|\leq M(a_{k}-a_{k+1})}$. That is, the magnitude of the partial sum of Bn, times a factor, is less than the upper bound of the partial sum Bn (a value M) times that same factor.

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

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

## Applications

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

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

Another corollary is that ${\displaystyle \sum _{n=1}^{\infty }a_{n}\sin n}$ converges whenever ${\displaystyle \{a_{n}\}}$ 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.

## Notes

1. ^ Démonstration d’un théorème d’Abel. Journal de mathématiques pures et appliquées 2nd series, tome 7 (1862), p. 253-255.

## 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.