# Lobachevskii criterion (for convergence)

From Encyclopedia of Mathematics

2010 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://www.encyclopediaofmath.org/index.php?title=Lobachevskii_criterion_(for_convergence)&oldid=39469

This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article