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Let $p>1$. I would like to have an estimate for the decay of the sequence $s_{n}=\sum_{k=n}^{\infty}k^{-p}$. Does anyone know of a bound of this type in the literature? Thanks!

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    The case where $p=2$ was examined in detail at this [MSE link](http://math.stackexchange.com/questions/685435/trying-to-get-a-bound-on-the-tail-of-the-series-for-zeta2).2014-03-27

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Look at the proof of the integral test of convergence for a sequence; we identify $s_n$ as upper and lower Riemann sums of integrals to get the bounds: $ \int_{n+1}^\infty x^{-p} \ dx \leq \sum_{k=n}^\infty \frac{1}{k^p} \leq \int_{n}^\infty x^{-p} \ dx. $ Evaluating the integrals, we then have $ \frac{1}{p-1} \frac{1}{(n+1)^{p}} \leq \sum_{k=n}^\infty \frac{1}{k^p} \leq \frac{1}{p-1} \frac{1}{n^p}. $

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    For finer-grained estimates based on refinements of the above idea, please see the [Euler-Maclaurin](http://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula) summation formula.2019-06-01