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Geometry Level 4

\[\mathfrak{T}_n=\cot^{-1} 2+\cot^{-1} 8+\cot^{-1} 18+\cdots \cot^{-1} 2n^2\]

\[\displaystyle \lim_{n\to\infty}\mathfrak{T}_n=\dfrac{\color{red}{\alpha }\pi}{\color{blue}{\gamma}}\] where \(\color{red}{\alpha}\) and \(\color{blue}{\gamma}\) are coprime. \[\Large \color{red}{\alpha} +4\color{blue}{\gamma}=\ ?\]

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