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 $\$ is a sequence of real numbers and $\$ a sequence of complex numbers satisfying
* $\$ is monotonically decreasing
heorem 1: Let an ≥ 0 be a decreasing sequence
* $\lim_a_n = 0$
* $\left, \sum^_b_n\\leq M$ for every positive integer ''N''
where ''M'' is some constant, then the series
$\sum^\infty_a_n b_n$
converges.

Proof

Let $S_n = \sum_^n a_k b_k$ and $B_n = \sum_^n b_k$.

From summation by parts, we have that $S_n = a_ B_ + \sum_^ B_k (a_k - a_)$. Since $B_n$ is bounded by ''M'' and $a_n \to 0$, the first of these terms approaches zero, $a_ B_ \to 0$ as $n\to\infty$.

We have, for each ''k'', $, B_k (a_k - a_), \leq M, a_k - a_,$. But, if $\$ is decreasing,
$\sum_^n M, a_k - a_, = \sum_^n M(a_k - a_) = M\sum_^n (a_k - a_),$
which is a telescoping sum, that equals $M(a_1 - a_)$ and therefore approaches $Ma_1$ as $n \to \infty$. Thus, $\sum_^\infty M(a_k - a_)$ converges. And, if $\$ is increasing,
$\sum_^n M, a_k - a_, = -\sum_^n M(a_k - a_) = -M\sum_^n (a_k - a_),$
which is again a telescoping sum, that equals $-M(a_1 - a_)$ and therefore approaches $-Ma_1$ as $n\to\infty$. Thus, again, $\sum_^\infty M(a_k - a_)$ converges.

So, the series $\sum_^\infty B_k(a_k - a_)$ converges, 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
$b_n = (-1)^n \Longrightarrow\left, \sum_^N b_n\ \leq 1.$

Another corollary is that $\sum_^\infty a_n \sin n$ converges whenever $\$ 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

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) .

External links

PlanetMath.org

{{Calculus topics

Convergence tests