D'Alembert criterion (convergence of series)

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If for a series of numbers,

there exists a number , , such that, from a certain term onwards, the inequality

is satisfied, the series converges absolutely; if, from a certain term onwards,

the series diverges. In particular, if the limit

exists, the series converges absolutely, and if

it diverges. For example, the series

converges absolutely for all complex , since

while the series diverges for all since


the series may converge or diverge; this condition is satisfied by the two series

the first series being convergent, while the second is divergent.

Established by J. d'Alembert (1768).


This criterion also goes by the name of ratio test, cf. [a1].


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