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

• ${\displaystyle a_{n+1}\leq a_{n}}$

• $\lim_{n \rightarrow \infty}a_n = 0$

• $\left|\sum^{N}_{n=1}b_n\right|\leq M$ for every positive integer N

where M is some constant, then the series

$\sum^{\infty}_{n=1}a_n b_n$

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

On the other hand, since the sequence $a_{n}$ is decreasing, $a_k - a_{k+1}$ is positive for all k, so $|B_k (a_k - a_{k+1})| \leq M(a_k - a_{k+1})$. That is, the magnitude of the partial sum of B_n, times a factor, is less than the upper bound of the partial sum B_n (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, $\sum_{k=0}^\infty |B_k(a_k - a_{k+1})|$ 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 $S_{n}$ 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 $\sum_{n=1}^\infty a_n \sin n$ 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.

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.

External links

• Proof at PlanetMath.org