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Calculus Level 5

\[\large \lim_{n \to \infty}\sum\limits_{m=1}^{n} \prod\limits_{k=1}^{m} \cos^{2} \! \left(\frac{π(2k-1)}{2(2m+1)}\right)=\frac{A}{B}\]

Find \(A+B\) where \(A\) and \(B\) are coprime positive integers.

Inspired by Otto's problem.

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