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Difference between revisions of "Dirichlet criterion (convergence of series)"

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If a sequence of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032820/d0328201.png" /> monotonically tends to zero, and the sequence of partial sums of the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032820/d0328202.png" /> is bounded (the terms of this series may also be complex), then the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d032/d032820/d0328203.png" /> converges. Established by P.G.L. Dirichlet [[#References|[1]]].
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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]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.G.L. Dirichlet,  ''J. de Math. (2)'' , '''7'''  (1862)  pp. 253–255</TD></TR></table>
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|valign="top"|{{Ref|Di}}|| P.G.L. Dirichlet,  ''J. de Math. (2)'' , '''7'''  (1862)  pp. 253–255
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See also [[Dedekind criterion (convergence of series)|Dedekind criterion (convergence of series)]].
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Revision as of 20:34, 9 December 2013

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 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, 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=13230
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article