Explanation

(c)
Let $S_{\infty} = \cot^{- 1}{2 + \cot^{- 1}{8 + \cot^{- 1}{18 + \cot^{- 1}{32 + ...}}}}$
$\therefore T_{n}\cot^{- 1}{2n^{2}}$
$= \tan^{- 1}\frac{1}{2n^{2}}$
$= \tan^{- 1}{\left( \frac{2}{4n^{2}} \right) = \tan^{- 1}\left( \frac{(2n + 1) - (2n - 1)}{1 + (2n + 1)(2n - 1)} \right)}\$
$\therefore S_{n} = \sum_{n = 1}^{\infty}{\{\tan^{- 1}{(2n + 1) - \tan^{- 1}{(2n - 1)\}}}}$
$= \tan^{- 1}{\infty - \tan^{- 1}1}$
$= \frac{\pi}{2} - \frac{\pi}{4} = \frac{\pi}{4}$