How find $\sum \limits _{k=1}^n \frac{1}{(k+1) \sqrt{k} + k \sqrt{k+1} }$

Question

How find sum $\sum \limits _{k=1}^n \frac{1}{(k+1) \sqrt{k} + k \sqrt{k+1} }$ ? Maybe there is a simple way.

Answer 1

We have $$\dfrac1{(k+1)\sqrt{k}+k\sqrt{k+1}} = \dfrac1{\sqrt{k}\sqrt{k+1}} \dfrac1{\sqrt{k+1}+\sqrt{k}} = \dfrac{\sqrt{k+1}-\sqrt{k}}{\sqrt{k}\sqrt{k+1}} = \dfrac1{\sqrt{k}} - \dfrac1{\sqrt{k+1}}$$ Hence, the summation telescopes to $1-\dfrac1{\sqrt{n+1}}$.