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-negative, non-increasing sequence $f(n)$ of real numbers, the series $\textstyle\sum_{n=1}^{\infty}f(n)$ converges if and only if the "condensed" series $\textstyle\sum_{n=0}^{\infty} 2^{n}f(2^{n})$ converges. Moreover, if they converge, the sum of the condensed series is no more than twice as large as the sum of the original.

单词	Cauchy condensation test
释义	Cauchy condensation test 英语百科 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-negative, non-increasing sequence $f(n)$ of real numbers, the series $\textstyle\sum_{n=1}^{\infty}f(n)$ converges if and only if the "condensed" series $\textstyle\sum_{n=0}^{\infty} 2^{n}f(2^{n})$ converges. Moreover, if they converge, the sum of the condensed series is no more than twice as large as the sum of the original.
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