View source for Lobachevskii criterion (for convergence)

{{MSC|40A05}}
{{TEX|done}}

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 test|Cauchy condensation test]].