Shifted integral representation of a p-series tail ($k_p=N-[(p-1){\sum }_{n=N}^{\infty }{n}^{-p}{]}^{-1/(p-1)}$)

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 thatIntegration 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 provesThe 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.