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{{MSC|40A05}}
{{TEX|done}}

''Gauss test''

A convergence criterion for a series of positive numbers $\sum_n a_n$, used by [[Gauss, Carl Friedrich|C. F. Gauss]] in 1812 to test the convergence of the [[hypergeometric series]]. The criterion states that, if the ratio
$\frac{a_n}{a_{n+1}}$ can be represented in the form
\begin{equation}\label{e:Gauss}
\frac{a_{n+1}}{a_n} = 1 - \frac{\alpha}{n} + \frac{\gamma_n}{n^\beta}\, ,
\end{equation}
where $\alpha$ and $\beta>1$ are constants and $\{\gamma_n\}$ is a bounded sequence, then the series converges if $\alpha> 1$ and diverges if $\alpha\leq 1$. Note that for \eqref{e:Gauss} to be valid it is necessary (but not sufficient) that the limit 
\[
\alpha = \lim_{n\to \infty} n \ln \left(\frac{a_n}{a_{n+1}}\right) = \lim_{n\to \infty} 
n \left(1-\frac{a_{n+1}}{a_n}\right) 
\]
exists. Gauss' criterion can therefore be naturally compared to [[Raabe criterion|Raabe's criterion]] and to [[Bertrand criterion|Bertrand's criterion]] and it is a simple case of a ''logarithmic convergence criterion'' (for a yet simpler one, see [[Logarithmic convergence criterion]]). 

====Comments====
The Gauss test is usually stated in the simpler form with $\beta =2$, cf. {{Cite|Kn}}, p. 297.

====References====
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|valign="top"|{{Ref|Kn}}|| K. Knopp,   "Theorie und   Anwendung der unendlichen Reihen" , Springer  (1964)  (English   translation: Blackie, 1951 &amp; Dover, reprint, 1990)
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