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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.
- for every positive integer N
where M is some constant, then the series
Let and .
From summation by parts, we have that .
Since is bounded by M and , the first of these terms approaches zero, as .
On the other hand, since the sequence is decreasing, is non-negative for all k, so . That is, the magnitude of the partial sum of , times a factor, is less than the upper bound of the partial sum (a value M) times that same factor.
But , which is a telescoping sum that equals and therefore approaches as . Thus, converges.
A particular case of Dirichlet's test is the more commonly used alternating series test for the case
Another corollary is that converges whenever is a decreasing sequence that tends to zero.
Moreover, Abel's test can be considered a special case of Dirichlet's test.
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.
- 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.