Difference between revisions of "Dirichlet criterion (convergence of series)"
From Encyclopedia of Mathematics
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| − | A criterion for the convergence of the series $\sum_n a_n b_n$, where $a_n$ are real numbers and $b_n$ are complex numbers, established by P.G.L. Dirichlet in {{Cite|Di}}. If a sequence of real numbers $a_n$ converges monotonically to zero, and the sequence of partial sums of the series $\sum_n b_n$ is bounded (the terms of this series may also be complex), then the series $\sum_n a_n b_n$ converges. The criterion is related to [[Dedekind criterion (convergence of series)|Dedekind's criterion]]. | + | A criterion for the convergence of the series $\sum_n a_n b_n$, where $a_n$ are real numbers and $b_n$ are complex numbers, established by P.G.L. Dirichlet and published posthumously in {{Cite|Di}}. If a sequence of real numbers $a_n$ converges monotonically to zero, and the sequence of partial sums of the series $\sum_n b_n$ is bounded (the terms of this series may also be complex), then the series $\sum_n a_n b_n$ converges. The criterion is related to [[Dedekind criterion (convergence of series)|Dedekind's criterion]]. |
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Latest revision as of 19:46, 16 January 2016
2020 Mathematics Subject Classification: Primary: 40A05 [MSN][ZBL]
A criterion for the convergence of the series $\sum_n a_n b_n$, where $a_n$ are real numbers and $b_n$ are complex numbers, established by P.G.L. Dirichlet and published posthumously in [Di]. If a sequence of real numbers $a_n$ converges monotonically to zero, and the sequence of partial sums of the series $\sum_n b_n$ is bounded (the terms of this series may also be complex), then the series $\sum_n a_n b_n$ converges. The criterion is related to Dedekind's criterion.
References
| [Di] | P.G.L. Dirichlet, "Démonstration d’un théorème d’Abel", J. de Math. (2) , 7 (1862) pp. 253–255 |
How to Cite This Entry:
Dirichlet criterion (convergence of series). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dirichlet_criterion_(convergence_of_series)&oldid=37553
Dirichlet criterion (convergence of series). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dirichlet_criterion_(convergence_of_series)&oldid=37553
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article