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Difference between revisions of "Lobachevskii criterion (for convergence)"

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A series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060010/l0600101.png" /> with positive terms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060010/l0600102.png" /> tending monotonically to zero converges or diverges according as the series
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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/l/l060/l060010/l0600103.png" /></td> </tr></table>
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A series $\sum_{n=1}^{\infty}a_n$ with positive terms $a_n$ tending monotonically to zero converges or diverges according as the series
 
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\begin{equation}
converges or diverges, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060010/l0600104.png" /> is the largest of the indices of the terms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060010/l0600105.png" /> that satisfy the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060010/l0600106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060010/l0600107.png" />.
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\sum_{m=0}^{\infty}\, p_m2^{-m}
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\end{equation}
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converges or diverges, where $p_m$ is the largest of the indices of the terms $a_n$ that satisfy the inequality $a_n\geq 2^{-m}$, $n=1,\dots,p_m$.
  
 
It was proposed by N.I. Lobachevskii in 1834–1836.
 
It was proposed by N.I. Lobachevskii in 1834–1836.

Revision as of 06:07, 14 December 2012


A series $\sum_{n=1}^{\infty}a_n$ with positive terms $a_n$ tending monotonically to zero converges or diverges according as the series \begin{equation} \sum_{m=0}^{\infty}\, p_m2^{-m} \end{equation} converges or diverges, where $p_m$ is the largest of the indices of the terms $a_n$ that satisfy the inequality $a_n\geq 2^{-m}$, $n=1,\dots,p_m$.

It was proposed by N.I. Lobachevskii in 1834–1836.

How to Cite This Entry:
Lobachevskii criterion (for convergence). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lobachevskii_criterion_(for_convergence)&oldid=17486
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article