Difference between revisions of "Lobachevskii criterion (for convergence)"
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| − | + | {{MSC|40A05}} | |
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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 | |
| + | \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. The criterion is essentially a "dual formulation" of what is commonly called the [[Cauchy condensation test]]. | |
| − | |||
| − | It was proposed by N.I. Lobachevskii in 1834–1836. | ||
Latest revision as of 16:29, 21 October 2016
2020 Mathematics Subject Classification: Primary: 40A05 [MSN][ZBL]
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. The criterion is essentially a "dual formulation" of what is commonly called the Cauchy condensation test.
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
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