= Shifted integral representation of a p-series tail
{title2=$k_p=N-[(p-1)\sum_{n=N}^\infty n^{-p}]^{-1/(p-1)}$}
Fix an integer $N>1$ and a real exponent $p>1$, and put $S_p=\sum_{n=N}^\infty n^{-p}$. There is a unique shift $k_p<N$ such that
$$
\int_N^\infty(x-k_p)^{-p}\,dx=S_p.
$$
Integration evaluates the left side as $(N-k_p)^{1-p}/(p-1)$, yielding the displayed formula. Comparing the decreasing function $x^{-p}$ with unit rectangles proves
$$
\frac{N^{1-p}}{p-1}<S_p<\frac{(N-1)^{1-p}}{p-1}.
$$
The integral is a strictly increasing function of its shift, so $0<k_p<1$. This is an exact reparametrization of a tail, not a claim that its shift is independent of $p$; interpolating shifts for different exponents is an additional approximation.
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