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Geometry Level 3

\[\large \lim \limits_{n\to \infty }\left[ \tan^{-1}\left( \frac{1}{2} \right) + \tan^{-1}\left( \frac{1}{8} \right) + \tan^{-1}\left( \frac{1}{18} \right) +\ldots+ \tan^{-1}\left( \frac{1}{2n^{2}} \right) \right] = \ ? \]

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