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

Put $p=k-1>0$. The preceding inequality bounds $n^{-k}<[(n-1{)}^{-p}-n^{-p}]/p$ for $n\ge 2$. For $N\ge 2$, summing the telescoping series givesThese nonnegative partial sums increase and are bounded, so the monotone convergence theorem for sequences proves convergence for every $k>1$. This is the telescoping bound for a p-series, an alternative to the integral test.