Sequences And Series

For any positive integer n, let S_n : (0, \(\infty\)) \(\to\) R be defined by \({S_n}(x) = \sum\nolimits_{k = 1}^n {{{\cot }^{ - 1}}\left( {{{1 + k(k + 1){x^2}} \over x}} \right)}\), where for any x \(\in\) R, \({\cot ^{ - 1}}(x) \in (0,\pi )\) and \({\tan ^{ - 1}}(x) \in \left( { - {\pi \over 2},{\pi \over 2}} \right)\). Then which of the following statements is (are) TRUE?