Cauchy condensation test

In mathematics, the Cauchy condensation test, named after Augustin-Louis Cauchy, is a standard convergence test for infinite series. For a non-increasing sequence ${\displaystyle f(n)}$ of non-negative real numbers, the series ${\displaystyle \displaystyle \sum \limits _{n=1}^{\infty }f(n)}$ converges if and only if the "condensed" series ${\displaystyle \displaystyle \sum \limits _{n=0}^{\infty }2^{n}f(2^{n})}$ converges. Moreover, if they converge, the sum of the condensed series is no more than twice as large as the sum of the original.