Dirichlet's test

In mathematics, Dirichlet's test is a method of testing for the convergence of a series. It is named after mathematician Johann Dirichlet who published it 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

• $a_n \geq a_{n+1} > 0$

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

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

where M is some constant, then the series

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

converges.

Proof

Let $S_n = \sum_{k=0}^n a_k b_k$ and $B_n = \sum_{k=0}^n b_k$. From summation by parts, we have that $S_n = a_{n+1} B_n + \sum_{k=0}^n B_k (a_k - a_{k+1})$. Since B_n is bounded and $a_n \rightarrow 0$, the first of these terms approaches zero. On the other hand, since the sequence a_n is decreasing, a_k − a_{k + 1} is positive for all k, so $|B_k (a_k - a_{k+1})| \leq M(a_k - a_{k+1})$. But $\sum_{k=0}^n M(a_k - a_{k+1}) = M(a_0 - a_{n+1}) \rightarrow Ma_0$, so the second term converges absolutely by the comparison test. Hence S_n converges.

Applications

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

$b_n = (-1)^n \Rightarrow\left|\sum_{n=1}^N b_n\right| \leq 1$.

Another corollary is that $\sum_{n=1}^\infty a_n \sin n$ converges whenever {a_n} is a decreasing sequence that tends to zero.

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