Necessary and sufficient conditions for the convergence of positive series

What is it about?

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].

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Why is it important?

It is a classic result of Mathematical Analysis that has important application in the theory of birth-and-death processes and Markov processes in general.

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