Kummer criterion

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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 .


[1] G.M. Fichtenholz, "Differential und Integralrechnung" , 2 , Deutsch. Verlag Wissenschaft. (1964)



[a1] E.D. Rainville, "Infinite series" , Macmillan (1967)
How to Cite This Entry:
Kummer criterion. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by E.G. Sobolevskaya (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article