Solution
= Solution
Put $p=k-1>0$. The preceding inequality bounds $n^{-k}<[ (n-1)^{-p}-n^{-p}]/p$ for $n\geq2$. For $N\geq2$, summing the <telescoping series> gives
$$
\sum_{n=1}^Nn^{-k}<1+\frac{1-N^{-(k-1)}}{k-1}<1+\frac1{k-1}.
$$
These nonnegative partial sums increase and are bounded, so the <monotone convergence theorem for sequences> proves \b[convergence for every $k>1$]. This is the <telescoping bound for a p-series>, an alternative to the <integral test>.