Rationalization of a square-root difference ($\sqrt{u}-\sqrt{v}=(u-v)/(\sqrt{u}+\sqrt{v})$)

For $u,v\ge 0$ with $u+v>0$, multiplication by the sum of the square roots givesThis identity exposes small differences hidden by cancellation. For example, $\sqrt{n^2+n}-n=1/(\sqrt{1+1/n}+1)\to 1/2$. It also converts normalized square-root increments into terms comparable with the harmonic series.