 Cauchy condensation test

If $(a_n)$ is a nonnegative decreasing sequence, then ${\sum }_{n\ge 1}{a}_n$ converges if and only if the condensed series ${\sum }_{k\ge 0}{2}^k{a}_{2^k}$ converges. Comparing each dyadic block with its first and last terms proves both directions.

The Cauchy condensation test is a convergence test used in the analysis of infinite series, particularly for series with non-negative terms. It provides a method to determine the convergence or divergence of a series based on the behavior of a related series.