Integrated Summation

Calculus Level 5

Consider the following sequence of variables defined as follows:

an=0π21cos2nx1cos2xdx\displaystyle a_n = \int\limits_0^\frac{\pi}{2} \frac{1- \cos 2nx }{1- \cos 2x } \text{d}x

If

S=n=11r=1narar+1ar+2ar+3\displaystyle S = \sum_{n=1}^\infty \frac{1}{\sum_{r=1}^n a_r a_{r+1} a_{r+2} a_{r+3}}

Evaluate 24π4S\displaystyle 24\pi ^4 \cdot S.

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