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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
The test states that if \{a_n\} is a sequence of real numbers and \{b_n\} a sequence of complex numbers satisfying



\lim_{n \rightarrow \infty}a_n = 0


\left\sum^{N}_{n=1}b_n\right\leq M for every positive integer N
where M is some constant, then the series

\sum^{\infty}_{n=1}a_n b_n
converges.
Proof
Let S_n = \sum_{k=0}^n a_k b_k and B_n = \sum_{k=0}^n b_k.
From summation by parts, we have that S_n = a_{n + 1} B_{n} + \sum_{k=0}^n B_k (a_k  a_{k+1}).
Since B_n is bounded by M and a_n \rightarrow 0, the first of these terms approaches zero, a_{n + 1}B_{n} \to 0 as n→∞.
On the other hand, since the sequence a_n is decreasing, a_k  a_{k+1} is positive for all k, so B_k (a_k  a_{k+1}) \leq M(a_k  a_{k+1}). That is, the magnitude of the partial sum of B_{n}, times a factor, is less than the upper bound of the partial sum B_{n} (a value M) times that same factor.
But \sum_{k=0}^n M(a_k  a_{k+1}) = M\sum_{k=0}^n (a_k  a_{k+1}), which is a telescoping series that equals M(a_0  a_{n+1}) and therefore approaches Ma_0 as n→∞. Thus, \sum_{k=0}^\infty M(a_k  a_{k+1}) converges.
In turn, \sum_{k=0}^\infty B_k(a_k  a_{k+1}) converges as well by the Direct Comparison test. The series \sum_{k=0}^\infty B_k(a_k  a_{k+1}) converges, as well, 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 \Rightarrow\left\sum_{n=1}^N b_n\right \leq 1.
Another corollary is that \sum_{n=1}^\infty a_n \sin n converges whenever \{a_n\} 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 nonnegative function, then the integral of fg is a convergent improper integral.
Notes

^ Démonstration d’un théorème d’Abel. Journal de mathématiques pures et appliquées 2nd series, tome 7 (1862), p. 253255.
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.1315) ISBN 082476949X.
External links
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