Find the sum `sum_(r =1)^(oo) tan^(-1) ((2(2r -1))/(4 + r^(2) (r^(2) -2r + 1)))`

Explanation

To find the sum

\[ S = \sum_{r=1}^{\infty} \tan^{-1} \left( \frac{2(2r - 1)}{4 + r^2(r^2 - 2r + 1)} \right), \]

we start by simplifying the expression inside the summation.

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