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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 ${\ extstyle \\sum \\limits _{n=1}^{\\infty }f(n)}$ converges if and only if the "condensed" series ${\ extstyle \\sum \\limits _{n=0}^{\\infty }2^{n}f(2^{n})}$ converges. Moreover, if they converge, the sum of the condensed series is no more than twice as large as the sum of the original.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Cauchy_condensation_test)