Gauss criterion

2020 Mathematics Subject Classification: Primary: 40A05 [MSN][ZBL]

Gauss test

A convergence criterion for a series of positive numbers $\sum_n a_n$, used by 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's criterion and to 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. [Kn], p. 297.

References

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