Cauchy condensation test - Imported Knowlpedia Content

Cauchy condensation test: Let $(a_n)$ be a nonincreasing sequence of real numbers with $a_n\ge 0$ . Then the series $\sum_{n=1}^\infty a_n$ converges if and only if the condensed series

\sum_{k=0}^\infty 2^k\,a_{2^k}

converges.

Remarks

This test is especially effective for borderline cases where comparison with $\sum 1/n^p$ is delicate; it is often used alongside the integral test and the comparison test.