Solution
= Solution
If $s\leq0$, 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 series
$$
\sum_{k=0}^{\infty}2^k(2^k)^{-s}
=\sum_{k=0}^{\infty}2^{k(1-s)}.
$$
This <geometric series> converges exactly when $1-s<0$. Hence the <p-series> criterion is
$$
\boxed{\sum_{n=1}^{\infty}n^{-s}\text{ converges}\iff s>1}.
$$