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.

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.

External links

• Proof at PlanetMath.org