Problem on convergence of infinite product

Question

Question : The infinite product $\prod_{n=2}^\infty(1-\frac{2}{n(n+1)})$ converges to what value ?

My attempt : I tried using partial fractions. The above product became

$\prod_{n=2}^\infty(1-\frac{2}{n} +\frac{2}{n+1})$

But it got more complicated.

I tried expanding terms

$\frac{2}{3}\times\frac{5}{6}\times\frac{9}{10}\times\frac{14}{15}...$

I can't find any pattern and simplify. I don't know how else to do. Kindly help me

Answer 1

HINT: $$1-\frac{2}{n(n+1)}=\frac{(n-1)(n+2)}{n(n+1)}$$Telescoping, anyone?