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''Gauss test''
 
''Gauss test''
  
A convergence criterion for a series of positive numbers
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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|hypergeomeric series]]. The criterion states that, if the ratio
 
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$\frac{a_n}{a_{n+1}}$ can be represented in the form
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043420/g0434201.png" /></td> </tr></table>
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\begin{equation}\label{e:Gauss}
 
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\frac{a_{n+1}}{a_n} = 1 - \frac{\alpha}{n} + \frac{\gamma_n}{n^\beta}\, ,
If the ratio <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043420/g0434202.png" /> can be represented in the form
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\end{equation}
 
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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  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043420/g0434203.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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\[
 
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\alpha = \lim_{n\to \infty} n \ln \left(\frac{a_n}{a_{n+1}}\right) = \lim_{n\to \infty}
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043420/g0434204.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043420/g0434205.png" /> are constants, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043420/g0434206.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043420/g0434207.png" /> is a bounded sequence, then the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043420/g0434208.png" /> converges if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043420/g0434209.png" /> and diverges if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043420/g04342010.png" />. For equation (*) to be valid it is necessary (but not sufficient) for the finite limit
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n \left(1-\frac{a_{n+1}}{a_n}\right)
 
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\]
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043420/g04342011.png" /></td> </tr></table>
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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]]).  
 
 
or
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043420/g04342012.png" /></td> </tr></table>
 
 
 
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|hypergeometric series]]. It is the simplest particular case of a [[Logarithmic convergence criterion|logarithmic convergence criterion]].
 
 
 
 
 
  
 
====Comments====
 
====Comments====
The criterion is usually stated in the simpler form with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043420/g04342013.png" />, cf. [[#References|[a1]]], p. 297.
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The Gauss test is usually stated in the simpler form with $\beta =2$, cf. {{Cite|Kn}}, p. 297.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Knopp,  "Theorie und Anwendung der unendlichen Reihen" , Springer  (1964) pp. 324 (English translation: Blackie, 1951 &amp; Dover, reprint, 1990)</TD></TR></table>
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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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Revision as of 10:54, 10 December 2013

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 hypergeomeric 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=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