  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
  $B_n\cdot \frac{a_n }{a_{n+1}} -B_{n+1}\geq 1 ,$ 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.
