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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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''of convergence of series $\sum_{n=1}^{\infty}a_n$ with positive numbers as terms''
  
 
If
 
If
 
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\begin{equation}
<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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B_n=\left[n\left(\frac{a_n}{a_{n+1}}-1\right)-1\right]\ln n
 
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\end{equation}
 
and if the limit (finite or infinite)
 
and if the limit (finite or infinite)
 
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\begin{equation}
<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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B = \lim_{n\to\infty}B_n
 
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\end{equation}
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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exists, then the series is convergent if $B>1$ and is divergent if $B<1$. Established by J. Bertrand.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.M. Fichtenholz,  "Differential und Integralrechnung" , '''1''' , Deutsch. Verlag Wissenschaft.  (1964)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.M. Fichtenholz,  "Differential und Integralrechnung" , '''1''' , Deutsch. Verlag Wissenschaft.  (1964)</TD></TR></table>

Revision as of 07:38, 13 December 2012


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

If \begin{equation} B_n=\left[n\left(\frac{a_n}{a_{n+1}}-1\right)-1\right]\ln n \end{equation} and if the limit (finite or infinite) \begin{equation} B = \lim_{n\to\infty}B_n \end{equation} exists, then the series is convergent if $B>1$ and is divergent if $B<1$. Established by J. Bertrand.

References

[1] 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