Product Under Summation (Part 2)

Calculus Level 5

\[\begin{eqnarray} && \text{S} = 1 - 5\left(\dfrac{1}{2}\right)^3 + 9 \left(\dfrac{1 \cdot 3}{2 \cdot 4}\right)^3 - 13\left(\dfrac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6}\right)^3 + \ldots \\ &&= 1 + \sum_{r=1}^{\infty} \left[ (-1)^r (4r+1) \prod_{k=1}^{r} \left(\dfrac{2k-1}{2k}\right)^3 \right] \end{eqnarray}\]

\(\text{S}\) can be represented as

\[\text{A} \cdot \pi^{\text{B}}\]

where \(\text{A}\) and \(\text{B}\) are integers.

Evaluate (\(3\text{A}+\text{B})^2\)


See also : Part 1, Part 3, Part 4

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