ratio test of d’Alembert

A lighter version of the ratio testMathworldPlanetmath is the

Ratio test of d’Alembert.  Let  a1+a2+  be a series with positive terms.

1.  If there exists a number q such that  0<q<1  and

an+1anqfor allnn0, (1)

then the series converges.

2.  If there exists a number n0 such that

an+1an 1for allnn0, (2)

then the series diverges.

Proof.1. By the condition (1), we have  an+1anq;  thus we get the estimations


Because  an0q+an0q2++an0qp+  is a convergentMathworldPlanetmathPlanetmath geometric seriesMathworldPlanetmath, those inequalitiesMathworldPlanetmath and the comparison testMathworldPlanetmath imply that the series


and as well the whole series  a1+a2+  is convergent.

2.  The condition (2) yields


and since  an0 is positive, the limit of an as n tends to infinity cannot be 0.  Hence the given series does not fulfil the necessary condition of convergence.

Example.  If the variable x in the power seriesMathworldPlanetmath


is distinct from zero, we have

|(n+1)!xn+1||n!xn|=(n+1)|x| 1for allnn0.

Then the series does not converge absolutely (http://planetmath.org/AbsoluteConvergence).  The known theorem of Abel says that the series diverges for all  x0.  It means that the radius of convergenceMathworldPlanetmath is 0.


  • 1 Л. Д. Кудрявцев: Математический анализ. I том.  Издательство  ‘‘Высшая школа’’. Москва (1970).
Title ratio test of d’Alembert
Canonical name RatioTestOfDAlembert
Date of creation 2013-03-22 19:12:28
Last modified on 2013-03-22 19:12:28
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Theorem
Classification msc 40A05
Related topic FiniteChangesInConvergentSeries