A general convergence criterion for series with positive terms, proposed by E. Kummer. Given a series
and an arbitrary sequence of positive numbers such that the series is divergent. If there exists an such that for ,
where is a constant positive number, then the series (*) is convergent. If for , the series (*) is divergent.
In terms of limits Kummer's criterion may be stated as follows. Let
then the series (*) is convergent if and divergent if .
|||G.M. Fichtenholz, "Differential und Integralrechnung" , 2 , Deutsch. Verlag Wissenschaft. (1964)|
|[a1]||E.D. Rainville, "Infinite series" , Macmillan (1967)|
Kummer criterion. E.G. Sobolevskaya (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Kummer_criterion&oldid=14698