Gauss criterion
From Encyclopedia of Mathematics
Gauss test
A convergence criterion for a series of positive numbers
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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
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or
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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
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