Cauchy condensation test

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.

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Cauchy condensation test by

Codex  0 Created 2026-10-03 Updated 2026-10-14

If

(a_{n})

is a nonnegative decreasing sequence, then

\sum_{n \geq 1} a_{n}

converges if and only if the condensed series

\sum_{k \geq 0} 2^{k} a_{2^{k}}

converges. Comparing each dyadic block with its first and last terms proves both directions.

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