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

A

$$S_{10}(x) = \frac{\pi}{2} - \tan^{-1}\left(\frac{1 + 11 x^2}{10 x}\right)$$, for all $$x > 0$$

B

$$\lim_{n \rightarrow \infty} \cot(S_n(x)) = x$$, for all x > 0

C

The equation $$S_3(x) = \frac{\pi}{4}$$ has a root in $$(0, \infty)$$

D

$$\tan(S_n(x)) \leq \frac{1}{2}$$, for all $$n \geq 1$$ and $$x > 0$$