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 formula_1 is a sequence of real numbers and formula_2 a sequence of complex numbers satisfying where "M" is some constant, then the series formula_6 converges. Proof. Let formula_7 and formula_8. From summation by parts, we have that formula_9. Since formula_10 is bounded by "M" and formula_11, the first of these terms  approaches zero, formula_12 as formula_13. We have, for each "k", formula_14. But, if formula_1 is decreasing, formula_16 which is a telescoping sum, that equals formula_17 and therefore approaches formula_18 as formula_19. Thus, formula_20 converges. And, if formula_1 is increasing, formula_22 which is again a telescoping sum, that equals formula_23 and therefore approaches formula_24 as formula_13. Thus, again, formula_20 converges. So, formula_27 converges as well by the direct comparison test. The series formula_28 converges, as well, by the absolute convergence test. Hence formula_29 converges. Applications. A particular case of Dirichlet's test is the more commonly used alternating series test for  the case formula_30 Another corollary is that formula_31 converges whenever formula_1 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.