D'Alembert criterion (convergence of series)

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)