Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2003 / Paper 1 / 10 / a / i / Solution

For $n\ge 6$ and $n-1<x<n$, monotonicity gives $(x+1{)}^{-2}<n^{-2}<x^{-2}$. Integrating over each unit interval and summing proves the integral test tail boundsThe inequalities remain strict because the very first rectangle already has a strictly positive difference from either bounding integral.