Question

state and prove cauchhyy conndensation test

Answer 1

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 {\displaystyle \displaystyle \sum \limits _{n=1}^{\infty }f(n)}converges if and only if the "condensed" series {\displaystyle \displaystyle \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.