Dirichlet criterion (convergence of series)
From Encyclopedia of Mathematics
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.
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=37554
Dirichlet criterion (convergence of series). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dirichlet_criterion_(convergence_of_series)&oldid=37554
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article