Test for series convergence
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 [ edit ] The test states that if ( a n ) {\displaystyle (a_{n})} is a sequence of real numbers and ( b n ) {\displaystyle (b_{n})} a sequence of complex numbers satisfying
( a n ) {\displaystyle (a_{n})} is monotonic lim n → ∞ a n = 0 {\displaystyle \lim _{n\to \infty }a_{n}=0} | ∑ n = 1 N b n | ≤ M {\displaystyle \left|\sum _{n=1}^{N}b_{n}\right|\leq M} for every positive integer N where M is some constant, then the series
∑ n = 1 ∞ a n b n {\displaystyle \sum _{n=1}^{\infty }a_{n}b_{n}} converges.
Let S n = ∑ k = 1 n a k b k {\textstyle S_{n}=\sum _{k=1}^{n}a_{k}b_{k}} and B n = ∑ k = 1 n b k {\textstyle B_{n}=\sum _{k=1}^{n}b_{k}} .
From summation by parts , we have that S n = a n + 1 B n + ∑ k = 1 n B k ( a k − a k + 1 ) {\textstyle S_{n}=a_{n+1}B_{n}+\sum _{k=1}^{n}B_{k}(a_{k}-a_{k+1})} . Since B n {\displaystyle B_{n}} is bounded by M and a n → 0 {\displaystyle a_{n}\to 0} , the first of these terms approaches zero, a n + 1 B n → 0 {\displaystyle a_{n+1}B_{n}\to 0} as n → ∞ {\displaystyle n\to \infty } .
We have, for each k , | B k ( a k − a k + 1 ) | ≤ M | a k − a k + 1 | {\displaystyle |B_{k}(a_{k}-a_{k+1})|\leq M|a_{k}-a_{k+1}|} .
Since ( a n ) {\displaystyle (a_{n})} is monotone, it is either decreasing or increasing:
If ( a n ) {\displaystyle (a_{n})} is decreasing, ∑ k = 1 n M | a k − a k + 1 | = ∑ k = 1 n M ( a k − a k + 1 ) = M ∑ k = 1 n ( a k − a k + 1 ) , {\displaystyle \sum _{k=1}^{n}M|a_{k}-a_{k+1}|=\sum _{k=1}^{n}M(a_{k}-a_{k+1})=M\sum _{k=1}^{n}(a_{k}-a_{k+1}),} which is a telescoping sum that equals M ( a 1 − a n + 1 ) {\displaystyle M(a_{1}-a_{n+1})} and therefore approaches M a 1 {\displaystyle Ma_{1}} as n → ∞ {\displaystyle n\to \infty } . Thus, ∑ k = 1 ∞ M ( a k − a k + 1 ) {\textstyle \sum _{k=1}^{\infty }M(a_{k}-a_{k+1})} converges. If ( a n ) {\displaystyle (a_{n})} is increasing, ∑ k = 1 n M | a k − a k + 1 | = − ∑ k = 1 n M ( a k − a k + 1 ) = − M ∑ k = 1 n ( a k − a k + 1 ) , {\displaystyle \sum _{k=1}^{n}M|a_{k}-a_{k+1}|=-\sum _{k=1}^{n}M(a_{k}-a_{k+1})=-M\sum _{k=1}^{n}(a_{k}-a_{k+1}),} which is again a telescoping sum that equals − M ( a 1 − a n + 1 ) {\displaystyle -M(a_{1}-a_{n+1})} and therefore approaches − M a 1 {\displaystyle -Ma_{1}} as n → ∞ {\displaystyle n\to \infty } . Thus, again, ∑ k = 1 ∞ M ( a k − a k + 1 ) {\textstyle \sum _{k=1}^{\infty }M(a_{k}-a_{k+1})} converges. So, the series ∑ k = 1 ∞ B k ( a k − a k + 1 ) {\textstyle \sum _{k=1}^{\infty }B_{k}(a_{k}-a_{k+1})} converges, by the absolute convergence test. Hence S n {\displaystyle S_{n}} converges.
Applications [ edit ] A particular case of Dirichlet's test is the more commonly used alternating series test for the case
b n = ( − 1 ) n ⟹ | ∑ n = 1 N b n | ≤ 1. {\displaystyle b_{n}=(-1)^{n}\Longrightarrow \left|\sum _{n=1}^{N}b_{n}\right|\leq 1.} Another corollary is that ∑ n = 1 ∞ a n sin n {\textstyle \sum _{n=1}^{\infty }a_{n}\sin n} converges whenever ( a n ) {\displaystyle (a_{n})} is a decreasing sequence that tends to zero. To see that ∑ n = 1 N sin n {\displaystyle \sum _{n=1}^{N}\sin n} is bounded, we can use the summation formula[2]
∑ n = 1 N sin n = ∑ n = 1 N e i n − e − i n 2 i = ∑ n = 1 N ( e i ) n − ∑ n = 1 N ( e − i ) n 2 i = sin 1 + sin N − sin ( N + 1 ) 2 − 2 cos 1 . {\displaystyle \sum _{n=1}^{N}\sin n=\sum _{n=1}^{N}{\frac {e^{in}-e^{-in}}{2i}}={\frac {\sum _{n=1}^{N}(e^{i})^{n}-\sum _{n=1}^{N}(e^{-i})^{n}}{2i}}={\frac {\sin 1+\sin N-\sin(N+1)}{2-2\cos 1}}.} Improper integrals [ edit ] 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.
References [ edit ] 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 [ edit ]