Necessary and sufficient conditions for the convergence of positive series

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), 452457] 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].