A series inspired by Satyajit

Calculus Level 5

\[\large{S = \frac{2}{3\times 4}+\frac{2 \times 4}{3\times 5 \times 6}+\frac{2\times 4 \times 6}{3\times 5\times 7\times 8}+ \ldots} \]

If \(S\) can be expressed as \(\dfrac{\pi^{A}}{B}-\dfrac{C}{D}\) where \(A,B,C,D\) are positive integers with \(C\) and \(D\) as coprime positive integers, evaluate \(A\times B\times C\times D\).


Inspiration.

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