A Minute Limit

Calculus Level 4

\[\lim_{ n \to \infty} n \left[ \dfrac{\tan^{-1} \sqrt{n -1}}{n^{2} - n + 1} + \dfrac{\tan^{-1} \sqrt{\dfrac{n-2}{2}}}{n^{2} - 2n + 4} + \dfrac{\tan^{-1} \sqrt{\dfrac{n-3}{3}}}{n^{2} - 3n + 9} + \ldots+ \dfrac{\tan^{-1} \sqrt{\dfrac{1}{n-1}}}{n^{2} - n + 1} \right] \]

If the limit above equals to \(\dfrac{\pi^{a}}{b\sqrt{c}} \) where\(c\) is square free, find \(a + b + c\).

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