Sum of series (telescoping)

Question

I have the following problem: $$\sum^{\infty}_{n=1}\dfrac{2}{\left(n+2\right)\sqrt{n}+n\sqrt{n+2}}$$ I should find the sum of this sequence. I tried to simplify but it does not work.

Answer 1

HINT:

$$\sqrt{n(n+2)}(\sqrt{n+2}+\sqrt n)=\sqrt{n(n+2)}\cdot\dfrac2{\sqrt{n+2}-\sqrt n}$$

$$\dfrac1{\sqrt{n(n+2)}(\sqrt{n+2}+\sqrt n)}=\dfrac{\sqrt{n+2}-\sqrt n}{2\sqrt{n(n+2)}}=f(n)-f(n+2)$$

where $f(m)=\dfrac1{2\sqrt m}$

Answer 2

Hint: $\frac{2}{(n+2)\sqrt{n}+(n)\sqrt{n+2}} $ can also be written as $\frac{(2)(\sqrt{n+2}-\sqrt{n})}{(\sqrt{(n)(n+2)})(\sqrt{n+2}+\sqrt{n}) (\sqrt{n+2}-\sqrt{n})}$ Now rationalize to get $\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+2}}$