D'Alembert criterion (convergence of series)
From Encyclopedia of Mathematics
If for a series of numbers,
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there exists a number ,
, such that, from a certain term onwards, the inequality
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is satisfied, the series converges absolutely; if, from a certain term onwards,
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the series diverges. In particular, if the limit
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exists, the series converges absolutely, and if
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it diverges. For example, the series
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converges absolutely for all complex , since
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while the series diverges for all
since
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If
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the series may converge or diverge; this condition is satisfied by the two series
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the first series being convergent, while the second is divergent.
Established by J. d'Alembert (1768).
Comments
This criterion also goes by the name of ratio test, cf. [a1].
References
[a1] | W. Rudin, "Principles of mathematical analysis" , McGraw-Hill (1953) |
How to Cite This Entry:
D'Alembert criterion (convergence of series). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=D%27Alembert_criterion_(convergence_of_series)&oldid=14634
D'Alembert criterion (convergence of series). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=D%27Alembert_criterion_(convergence_of_series)&oldid=14634
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article