Английская Википедия:Dirichlet's test
Шаблон:Short description Шаблон:Calculus
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 <math>(a_n)</math> is a sequence of real numbers and <math>(b_n)</math> a sequence of complex numbers satisfying
- <math>(a_n)</math> is monotonic
- <math>\lim_{n \to \infty}a_n = 0</math>
- <math>\left|\sum_{n=1}^{N}b_n\right| \leq M</math> for every positive integer N
where M is some constant, then the series
- <math display="block">\sum_{n=1}^{\infty} a_n b_n</math>
converges.
Proof
Let <math display="inline">S_n = \sum_{k=1}^n a_k b_k</math> and <math display="inline">B_n = \sum_{k=1}^n b_k</math>.
From summation by parts, we have that <math display="inline">S_n = a_n B_n + \sum_{k=1}^{n-1} B_k (a_k - a_{k+1})</math>. Since <math>B_n</math> is bounded by M and <math>a_n \to 0</math>, the first of these terms approaches zero, <math>a_n B_n \to 0</math> as <math>n\to\infty</math>.
We have, for each k, <math>|B_k (a_k - a_{k+1})| \leq M|a_k - a_{k+1}|</math>.
Since <math>(a_n)</math> is monotone, it is either decreasing or increasing:
- If <math>(a_n)</math> is decreasing, <math display="block"> \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}),</math> which is a telescoping sum that equals <math>M(a_1 - a_{n+1})</math> and therefore approaches <math>Ma_1</math> as <math>n \to \infty</math>. Thus, <math display="inline"> \sum_{k=1}^\infty M(a_k - a_{k+1})</math> converges.
- If <math>(a_n)</math> is increasing, <math display="block"> \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}),</math> which is again a telescoping sum that equals <math>-M(a_1 - a_{n+1})</math> and therefore approaches <math>-Ma_1</math> as <math>n\to\infty</math>. Thus, again, <math display="inline"> \sum_{k=1}^\infty M(a_k - a_{k+1})</math> converges.
So, the series <math display="inline"> \sum_{k=1}^\infty B_k(a_k - a_{k+1})</math> converges, by the absolute convergence test. Hence <math>S_n</math> converges.
Applications
A particular case of Dirichlet's test is the more commonly used alternating series test for the case <math display="block">b_n = (-1)^n \Longrightarrow\left|\sum_{n=1}^N b_n\right| \leq 1.</math>
Another corollary is that <math display="inline"> \sum_{n=1}^\infty a_n \sin n </math> converges whenever <math>(a_n)</math> is a decreasing sequence that tends to zero. To see that <math> \sum_{n=1}^N \sin n </math> is bounded, we can use the summation formula[2] <math display="block">\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}.</math>
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.
Notes
- ↑ Démonstration d’un théorème d’Abel. Journal de mathématiques pures et appliquées 2nd series, tome 7 (1862), pp. 253–255 Шаблон:Webarchive.
- ↑ Шаблон:Cite web
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) Шаблон:ISBN.
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