Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2006 / Paper 2 / 10 / d / Solution

Subtract the right side from the left and write $g(x)=1-(1+x{)}^{-p}-px(1+x{)}^{-p-1}$. Then $g(0)=0$ and $g^{\prime }(x)=p(p+1)x(1+x{)}^{-p-2}>0$ for $x>0$, so the strict inequality follows. Set $x=1/n$ and multiply by $n^{-p}>0$ to obtainThis decreasing-power difference supplies a telescoping bound for a p-series.