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Difference between revisions of "Kummer criterion"

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A general convergence criterion for series with positive terms, proposed by E. Kummer. Given a series
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{{MSC|40A05}}
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{{TEX|done}}
  
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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/k/k055/k055950/k0559501.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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A general convergence criterion for series with positive terms, proposed by E. Kummer. Let
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\begin{equation}\label{e:series}
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\sum_n a_n
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\end{equation}
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be a series of positive numbers and $\{c_n\}$ a sequence of positive numbers. If there are $\delta >0$ and $N$ such that
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\[
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K_n := c_n \frac{a_n}{a_{n+1}} - c_{n+1} \geq \delta \qquad \forall n\geq N\, ,
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\]
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then \eqref{e:series} converges. If the series $\sum_n (c_n)^{-1}$ diverges and there is $N$ such that $K_n \leq 0$ for all $n\geq N$, then \eqref{e:series} diverges.
  
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and an arbitrary sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055950/k0559502.png" /> of positive numbers such that the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055950/k0559503.png" /> is divergent. If there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055950/k0559504.png" /> such that for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055950/k0559505.png" />,
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An obvious corollary is that, when the limit
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\[
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055950/k0559506.png" /></td> </tr></table>
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K := \lim_{n\to \infty} K_n
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\]
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055950/k0559507.png" /> is a constant positive number, then the series (*) is convergent. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055950/k0559508.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055950/k0559509.png" />, the series (*) is divergent.
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exists we have:
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* if $K>0$ \eqref{e:series} converges
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In terms of limits Kummer's criterion may be stated as follows. Let
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* if $K<0$ and $\sum_n (c_n)^{-1}$ diverges, then \eqref{e:series} diverges.
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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/k/k055/k055950/k05595010.png" /></td> </tr></table>
 
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then the series (*) is convergent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055950/k05595011.png" /> and divergent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055950/k05595012.png" />.
 
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====References====
 
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G.M. Fichtenholz,   "Differential und Integralrechnung" , '''2''' , Deutsch. Verlag Wissenschaft. (1964)</TD></TR></table>
 
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====Comments====
 
  
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.D. Rainville,  "Infinite series" , Macmillan  (1967)</TD></TR></table>
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{|
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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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|-
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|valign="top"|{{Ref|Ra}}|| E.D. Rainville,  "Infinite series" , Macmillan  (1967)
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|-
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|}

Latest revision as of 11:25, 10 December 2013

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

A general convergence criterion for series with positive terms, proposed by E. Kummer. Let \begin{equation}\label{e:series} \sum_n a_n \end{equation} be a series of positive numbers and $\{c_n\}$ a sequence of positive numbers. If there are $\delta >0$ and $N$ such that \[ K_n := c_n \frac{a_n}{a_{n+1}} - c_{n+1} \geq \delta \qquad \forall n\geq N\, , \] then \eqref{e:series} converges. If the series $\sum_n (c_n)^{-1}$ diverges and there is $N$ such that $K_n \leq 0$ for all $n\geq N$, then \eqref{e:series} diverges.

An obvious corollary is that, when the limit \[ K := \lim_{n\to \infty} K_n \] exists we have:

  • if $K>0$ \eqref{e:series} converges
  • if $K<0$ and $\sum_n (c_n)^{-1}$ diverges, then \eqref{e:series} diverges.


References

[Fi] G.M. Fichtenholz, "Differential und Integralrechnung" , 1 , Deutsch. Verlag Wissenschaft. (1964)
[Ra] E.D. Rainville, "Infinite series" , Macmillan (1967)
How to Cite This Entry:
Kummer criterion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kummer_criterion&oldid=14698
This article was adapted from an original article by E.G. Sobolevskaya (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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