Infinite Summation + Infinite Products

Calculus Level 5

\[\large\displaystyle\sum_{n=1}^{\infty}\prod_{k=n+1}^{\infty}\left(1-\left(\dfrac{n}k\right)^2\right)\] Let \(S\) denote the value of series above. If \(S=\dfrac{A}{B}+\dfrac{C\pi}{D\sqrt{B}}\) where \(A,B,C\) and \(D\) are coprime positive integers. Find the value of \(A+B+C+D\).

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