A Short and Unified Proof of Kummer's Test

Abstract

Kummer's test from 1835 states that the positive series ${\sum }_{n=1}^{\infty }{a}_n$ is convergent if and only if there is a sequence $\{B_n{\}}_1^{\infty }$ of positive numbers such that $B_n\cdot \frac{a_n}{a_{n+1}}-B_{n+1}\ge 1,$ for all sufficiently large $n$. We present an exact analysis and a short and unified proof of Kummer's test. The test has been applied to differential equations and studied in mathematical philosophy.

4 pages