Product under Summation (Part 3)

Calculus Level 5

\[\begin{align*} \text{S} &= 1 + \dfrac{1}{3^2}\left( \dfrac{1}{2}\right) +\dfrac{1}{5^2} \left( \dfrac{1\cdot 3}{2\cdot 4} \right) + \dfrac{1}{7^2}\left( \dfrac{1\cdot 3 \cdot 5}{2\cdot 4 \cdot 6} \right) + \ldots \\ & = 1 + \sum_{r=1}^{\infty} \left[ \dfrac{1}{(2r+1)^2}\prod_{k=1}^{r}\left(\dfrac{2k-1}{2k}\right) \right] \end{align*} \]

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

\[\dfrac{{\text{A} \pi}\ln (\text{B})}{\text{C}}\]

where \({\text{A}}\), \({\text{B}}\) and \({\text{C}}\) are positive integers, \({\text{A}}\) and \({\text{C}}\) are coprime and \({\text{B}}\) is a prime number.

Evaluate \({\text{A}}+{\text{B}}+{\text{C}}\).


See also : Part 1, Part 2, Part 4

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