If $A_i (i =1,2,3..........,n)$ are $n$ independent events with $ P(A_i)=\large\frac{1}{1+i}$ for each i, then the probability that none of $A_i$ occurs is :

Question

\[\begin {array} {1 1} (1)\;\frac{n-1}{n+1} & \quad (2)\;\frac{n}{n+1} \\ (3)\;\frac{n}{n+2} & \quad (4)\;\frac{1}{n+1} \end {array}\]