Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2004 / Paper 2 / 10 / b / i / Solution

The shifted integral representation of a p-series tail evaluates the defining integral asIt is strictly increasing in $k$, since $I_p^{\prime }(k)=(5-k{)}^{-p}>0$. From the bounds just proved with $a=5$, $I_3(0)<{\sum }_{n=5}^{\infty }{n}^{-3}<I_3(1)$. Since the middle expression equals $I_3(k_3)$, monotonicity proves