# D'Alembert criterion (convergence of series)

From Encyclopedia of Mathematics

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

If

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).

#### 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

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