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Difference between revisions of "Bertrand criterion"

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''of convergence of series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015780/b0157801.png" /> with positive numbers as terms''
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{{MSC|40A05}}
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{{TEX|done}}
  
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If
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''for convergence of series $\sum_{n=1}^{\infty} a_n$ of positive numbers''
  
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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/b/b015/b015780/b0157802.png" /></td> </tr></table>
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A onvergence criterion for series $\sum_n a_n$ of positive real numbers, established by [[Joseph Bertrand|J. Bertrand]]. Assume that the limit
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\begin{equation}
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and if the limit (finite or infinite)
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B = \lim_{n\to\infty} \left[n\left(\frac{a_n}{a_{n+1}}-1\right)-1\right]\ln n\,
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\end{equation}
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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/b/b015/b015780/b0157803.png" /></td> </tr></table>
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exists. If $B>1$ then the series converges and if $B<1$, then the series diverges. If the limit is $1$, then the convergence cannot be decided, as it is witnessed by the examples
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\[
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exists, then the series is convergent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015780/b0157804.png" /> and is divergent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015780/b0157805.png" />. Established by J. Bertrand.
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\sum_{n\geq 2} \frac{1}{n \log n}
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\]
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(which diverges) and
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\[
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\sum_{n\geq 3} \frac{1}{n \log n (\log \log n)^2}\,
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\]
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(which converges).
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.M. Fichtenholz,  "Differential und Integralrechnung" , '''1''' , Deutsch. Verlag Wissenschaft.  (1964)</TD></TR></table>
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{|
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|valign="top"|{{Ref|Fi}}|| G.M. Fichtenholz,  "Differential und Integralrechnung" , '''1''' , Deutsch. Verlag Wissenschaft.  (1964)
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Latest revision as of 10:25, 16 March 2023

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

for convergence of series $\sum_{n=1}^{\infty} a_n$ of positive numbers

A onvergence criterion for series $\sum_n a_n$ of positive real numbers, established by J. Bertrand. Assume that the limit \begin{equation} B = \lim_{n\to\infty} \left[n\left(\frac{a_n}{a_{n+1}}-1\right)-1\right]\ln n\, \end{equation} exists. If $B>1$ then the series converges and if $B<1$, then the series diverges. If the limit is $1$, then the convergence cannot be decided, as it is witnessed by the examples \[ \sum_{n\geq 2} \frac{1}{n \log n} \] (which diverges) and \[ \sum_{n\geq 3} \frac{1}{n \log n (\log \log n)^2}\, \] (which converges).

References

[Fi] G.M. Fichtenholz, "Differential und Integralrechnung" , 1 , Deutsch. Verlag Wissenschaft. (1964)
How to Cite This Entry:
Bertrand criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bertrand_criterion&oldid=16104
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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