Product under Summation (Part 4)

Calculus Level 5

\[\begin{eqnarray} S &=& 1 + 9\left(\dfrac{1}{4}\right)^4 + 17\left(\dfrac{1\cdot 5}{4\cdot 8}\right)^4 + 25\left(\dfrac{1\cdot 5\cdot 9}{4\cdot 8\cdot 12}\right)^4 + \ldots \\ &=& 1 + \sum_{r=1}^{\infty} \left[ (8r+1) \prod_{k=1}^{r} \left(\dfrac{4k-3}{4k}\right)^4 \right] \end{eqnarray}\]

Given that \(S\) can be represented as \(\dfrac{A^{\frac{3}{2}} \pi^{-\frac{B}{C}}}{\Gamma^2\left(\dfrac{D}{E}\right)}\) where \(A,B,C,D\) and \(E\) are positive integers with \(\gcd (B,C) = \gcd(D,E) = 1\).

Evaluate \(A+B+C+D+E\)


See Also : Part 1, Part 2, Part 3

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