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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:

If $s\le 0$, the terms $n^{-s}$ do not tend to zero, so the series diverges by the term test for divergence. If $s>0$, the terms form a nonnegative decreasing sequence, and the Cauchy condensation test gives the condensed seriesThis geometric series converges exactly when $1-s<0$. Hence the p-series criterion is