Difference between revisions of "Gauss criterion"
From Encyclopedia of Mathematics
(Importing text file) |
m |
||
| (2 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
| + | {{MSC|40A05}} | ||
| + | {{TEX|done}} | ||
| + | |||
''Gauss test'' | ''Gauss test'' | ||
| − | A convergence criterion for a series of positive numbers | + | 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 | |
| − | + | \[ | |
| − | + | \alpha = \lim_{n\to \infty} n \ln \left(\frac{a_n}{a_{n+1}}\right) = \lim_{n\to \infty} | |
| − | where | + | n \left(1-\frac{a_{n+1}}{a_n}\right) |
| − | + | \] | |
| − | + | 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==== | ====Comments==== | ||
| − | The | + | The Gauss test is usually stated in the simpler form with $\beta =2$, cf. {{Cite|Kn}}, p. 297. |
====References==== | ====References==== | ||
| − | + | {| | |
| + | |- | ||
| + | |valign="top"|{{Ref|Kn}}|| K. Knopp, "Theorie und Anwendung der unendlichen Reihen" , Springer (1964) (English translation: Blackie, 1951 & Dover, reprint, 1990) | ||
| + | |- | ||
| + | |} | ||
Latest revision as of 20:01, 21 March 2023
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) |
How to Cite This Entry:
Gauss criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gauss_criterion&oldid=17936
Gauss criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Gauss_criterion&oldid=17936
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article