  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].
