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Dirichlet's test

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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.[1]

Statement

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

  • a_n \geq a_{n+1}
  • \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 Bn, times a factor, is less than the upper bound of the partial sum Bn (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 non-negative function, then the integral of fg is a convergent improper integral.

Notes

  1. ^ Démonstration d’un théorème d’Abel. Journal de mathématiques pures et appliquées 2nd series, tome 7 (1862), p. 253-255.

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) ISBN 0-8247-6949-X.

External links

  • Proof at PlanetMath.org
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