# Extension of the Bertrand-De Morgan test and its application

## TL;DR

This paper presents a simplified proof of the extended Bertrand-De Morgan test for series convergence and demonstrates its application in birth-and-death processes and random walks.

## Contribution

It offers a straightforward proof of the extended test and applies it to stochastic process analysis, enhancing understanding and utility.

## Key findings

- Simplified proof of the extended Bertrand-De Morgan test
- Application of the test in birth-and-death processes
- Application of the test in random walks

## Abstract

We provide a simple proof for the extended Bertrand-De Morgan test that was earlier studied in [F. \v{D}uri\v{s}, Infinite Series: Convergence tests. Bachelor thesis, 2009] and [A.A. Tabatabai Adnani, A. Reza and M. Morovati, J. Lin. Topol. Algebra, 2 (3), pp. 141--147 (2013)] and demonstrate an application of that test in the theory of birth-and-death processes and random walks.