Gauss criterion

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Gauss test

A convergence criterion for a series of positive numbers

If the ratio can be represented in the form


where and are constants, and is a bounded sequence, then the series converges if and diverges if . For equation (*) to be valid it is necessary (but not sufficient) for the finite limit


to exist. Gauss' criterion was historically (1812) one of the first general criteria for convergence of a series of numbers. It was employed by C.F. Gauss to test the convergence of the hypergeometric series. It is the simplest particular case of a logarithmic convergence criterion.


The criterion is usually stated in the simpler form with , cf. [a1], p. 297.


[a1] K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) pp. 324 (English translation: Blackie, 1951 & Dover, reprint, 1990)
How to Cite This Entry:
Gauss criterion. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article