Difference between revisions of "Lobachevskii criterion (for convergence)"
From Encyclopedia of Mathematics
m (TeX encoding is done) |
|||
| Line 1: | Line 1: | ||
| + | {{MSC|40A05}} | ||
{{TEX|done}} | {{TEX|done}} | ||
| Line 7: | Line 8: | ||
converges or diverges, where $p_m$ is the largest of the indices of the terms $a_n$ that satisfy the inequality $a_n\geq 2^{-m}$, $n=1,\dots,p_m$. | 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. The criterion is essentially a "dual formulation" of what is commonly called the [[Cauchy test|Cauchy condensation test]]. |
Revision as of 11:04, 10 December 2013
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=30924
Lobachevskii criterion (for convergence). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lobachevskii_criterion_(for_convergence)&oldid=30924
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article