Lobachevskii criterion (for convergence)

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.