In mathematics, Dirichlet's test is a method of testing for the convergence of a series that is especially useful for proving conditional convergence. 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 (a_n) is a monotonic sequence of real numbers with and (b_n) is a sequence of real numbers or complex numbers with bounded partial sums, then the series converges.

Proof

Let and. From summation by parts, we have that. Since the magnitudes of the partial sums B_n are bounded by some M and a_n \to 0 as n\to\infty, the first of these terms approaches zero: as n\to\infty. Furthermore, for each k,. Since (a_n) is monotone, it is either decreasing or increasing:

<ul><li> If (a_n) is decreasing, which is a [telescoping sum](https://bliptext.com/articles/telescoping-sum) that equals and therefore approaches Ma_1 as. Thus, converges. <li> If (a_n) is increasing, which is again a telescoping sum that equals and therefore approaches -Ma_1 as n\to\infty. Thus, again, converges. </ul> So, the series converges by the [direct comparison test](https://bliptext.com/articles/direct-comparison-test) to. Hence S_n 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 (a_n) 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.