Kummer criterion

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 .

References

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

References

[a1]	E.D. Rainville, "Infinite series" , Macmillan (1967)