Namespaces
Variants
Actions

Gauss criterion

From Encyclopedia of Mathematics
Revision as of 17:23, 7 February 2011 by 127.0.0.1 (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

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)
How to Cite This Entry:
Gauss criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gauss_criterion&oldid=30923
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article