# Kummer criterion

From Encyclopedia of Mathematics

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) |

#### Comments

#### References

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

**How to Cite This Entry:**

Kummer criterion.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Kummer_criterion&oldid=14698

This article was adapted from an original article by E.G. Sobolevskaya (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article