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
editThe test states that if is a sequence of real numbers and a sequence of complex numbers satisfying
where M is some constant, then the series
converges.
Proof
editLet and .
From summation by parts, we have that . Since is bounded by M and , the first of these terms approaches zero, as .
We have, for each k, .
Since is monotone, it is either decreasing or increasing:
- If is decreasing, which is a telescoping sum that equals and therefore approaches as . Thus, converges.
- If is increasing, which is again a telescoping sum that equals and therefore approaches as . Thus, again, converges.
So, the series converges, by the absolute convergence test. Hence converges.
Applications
editA 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. To see that is bounded, we can use the summation formula[2]
Improper integrals
editAn 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 non-negative monotonically decreasing function, then the integral of fg is a convergent improper integral.
Notes
edit- ^ Démonstration d’un théorème d’Abel. Journal de mathématiques pures et appliquées 2nd series, tome 7 (1862), pp. 253–255 Archived 2011-07-21 at the Wayback Machine. See also [1].
- ^ "Where does the sum of $\sin(n)$ formula come from?".
References
edit- 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.