Necessary and sufficient conditions for the convergence of positive series

TL;DR

This paper establishes new necessary and sufficient criteria for the convergence of positive series, extending classical tests and applying these results to analyze recurrence and transience in birth-and-death processes.

Contribution

It introduces novel conditions that unify and extend existing convergence tests for positive series, with implications for stochastic process analysis.

Findings

New criteria for positive series convergence

Extension of recurrence and transience conditions

Enhanced understanding of birth-and-death processes

Abstract

We provide new necessary and sufficient conditions for the convergence of positive series developing Bertran-De Morgan and Cauchy type tests given in [M. Martin, Bull. Amer. Math. Soc. 47(1941), 452-457] and [L. Bourchtein et al, Int. J. Math. Anal. 6(2012), 1847-1869]. The obtained result enables us to extend the known conditions for recurrence and transience of birth-and-death processes given in [V. M. Abramov, Amer. Math. Monthly 127(2020) 444-448].