= Telescoping bound for a p-series
{title2=$\sum_{n=1}^Nn^{-k}\leq1+[1-N^{-(k-1)}]/(k-1),\quad k>1$}
For $p>0$ and $n\geq2$, integrating the decreasing derivative magnitude of $x^{-p}$ gives $(n-1)^{-p}-n^{-p}=p\int_{n-1}^nx^{-p-1}dx>pn^{-p-1}$. Put $p=k-1$ and sum this <telescoping series> to obtain the bound in the title, strict for $N>1$. The partial sums are increasing and bounded, proving convergence of the <p-series> when $k>1$. This gives a direct <comparison test for series> proof and a finite upper bound without evaluating the sum.
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