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 monotonic
 
  for every positive integer N
where M is some constant, then the series

converges. Proof 

Let  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:
<li>
If  is decreasing,

which is a telescoping sum that equals  and therefore approaches  as . Thus,  converges. <li>
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 

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. To see that

is bounded, we can use the summation formula

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 non-negative monotonically decreasing 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

Convergence tests