Product under Summation (Part 1)

Calculus Level 5

\[\begin{eqnarray} && \text{S} = 1 + \dfrac{1}{4} + \dfrac {1}{16}\left(\dfrac{5}{4} \right) + \dfrac {1}{36}\left(\dfrac{5 \cdot 17}{4 \cdot 16} \right) + \dfrac{1}{64} \left(\dfrac{5 \cdot 17 \cdot 37 }{4 \cdot 16 \cdot 36} \right) + \cdots \\ &&= 1 + \dfrac14 +\sum_{r=1}^\infty \left [ \dfrac1{4(r+1)^2} \prod_{k=1}^r \left( \dfrac1{4k^2} + 1 \right) \right ] \end{eqnarray} \]

\(\text{S}\) can be represented as \[ \dfrac{\text{A}}{\pi^{\text{B}}} \sinh \left( \dfrac{\text{C}\pi ^{\text{D}}}{\text{E}} \right) \] where \(\text{A},\text{B},\text{C},\text{D}\) and \(\text{E}\) are positive integers with \(\text{C},\text{E}\) coprime. Compute \(\text{A}+\text{B}+\text{C}+\text{D}-\text{E}\).


See also : Part 2, Part 3, Part 4

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