A Short and Unified Proof of Kummer's Test

Kummer’s test from 1835 states that the positive series $\sum_{n=1}^\infty a_n$isconvergentifandonlyifthereisasequence${ B_n}_1^\infty$ 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.