Dirichlet's test

Part of a series of articles about

Definitions
Derivative (generalizations) Differential infinitesimal of a function total
Concepts
Differentiation notation Second derivative Third derivative Change of variables Implicit differentiation Related rates Taylor's theorem
Rules and identities
Sum Product Chain Power Quotient General Leibniz Faà di Bruno's formula

Integral

Definitions
Lists of integrals
Antiderivative Integral (improper) Riemann integral Lebesgue integration Contour integration
Integration by
Parts Discs Cylindrical shells Substitution (trigonometric) Partial fractions Order Reduction formulae

Series

Convergence tests
Geometric (arithmetico-geometric) Harmonic Alternating Power Binomial Taylor
Summand limit (term test) Ratio Root Integral Direct comparison Limit comparison Alternating series Cauchy condensation Dirichlet Abel

Vector

Theorems
Gradient Divergence Curl Laplacian Directional derivative Identities
Divergence Gradient Green's Kelvin–Stokes

Multivariable

Formalisms
Matrix Tensor Exterior Geometric
Definitions
Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian Hessian matrix

Specialized

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

$a_{n+1}\leq a_{n}$

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

$\left|\sum^{N}_{n=1}b_n\right|\leq M$ for every positive integer N

where M is some constant, then the series

\sum^{\infty}_{n=1}a_n b_n

converges.

Proof

Let $S_{n}=\sum _{k=1}^{n}a_{k}b_{k}$ and $B_{n}=\sum _{k=1}^{n}b_{k}$ .

From summation by parts, we have that $S_{n}=a_{n+1}B_{n}+\sum _{k=1}^{n}B_{k}(a_{k}-a_{k+1})$ .

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 $\sum _{k=1}^{n}M(a_{k}-a_{k+1})=M\sum _{k=1}^{n}(a_{k}-a_{k+1})$ , which is a telescoping series that equals $M(a_{1}-a_{n+1})$ and therefore approaches $Ma_{1}$ as n→∞. Thus, $\sum _{k=1}^{\infty }M(a_{k}-a_{k+1})$ converges.

In turn, $\sum_{k=0}^\infty |B_k(a_k - a_{k+1})|$ converges as well by the Direct comparison test. The series $\sum _{k=1}^{\infty }B_{k}(a_{k}-a_{k+1})$ 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

↑ 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

This article is issued from Wikipedia - version of the 11/23/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.