Gauss criterion

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

or

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.

Comments

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

References

[a1]	K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) pp. 324 (English translation: Blackie, 1951 & Dover, reprint, 1990)