Past exam of the mathematics course of the University of Cambridge / 2017 / ia / Paper 1 / 11F / a / Solution

For each $k\ge 0$, monotonicity gives the dyadic-block boundsSumming the upper bounds proves convergence of $\sum x_n$ whenever $\sum 2^k{x}_{2^k}$ converges. Summing the lower bounds gives, up to the first term,so convergence of $\sum x_n$ forces convergence of the condensed series. This is the Cauchy condensation test: