\prod_{k=2}^n \frac{(k-1)}{(k+1)}=\frac{1}{3}\cdot\frac{2}{4}\cdot\frac{3}{5}\cdot\ldots\cdot\frac{n-3}{n-1}\cdot\frac{n-2}{n}\cdot\frac{n-1}{n+1}=\frac{2}{n(n+1)}