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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.

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.