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For any positive integer \( \mathrm{n} \), let \( S_{\mathrm{n}}:(0, \infty) \rightarrow \mathbb{R} \) be defined by \( S_{n}(x)=\sum_{k=1}^{n} \cot ^{-1}\left(\frac{1+k(k+1) x^{2}}{x}\right) \)
\( \mathrm{P} \)
W)
Where for any \( x \in \mathbb{R}, \cot ^{-1} x \in(0, \pi) \) and \( \tan ^{-1}(x) \in \)
\( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \). Then which of the following statements is (are)
TRUE?
[JEE(Advanced)-2021]
(a) \( S_{10}(x)=\frac{\pi}{2}-\tan ^{-1}\left(\frac{1+11 x^{2}}{10 x}\right) \), for all \( x0 \)
(b) \( \lim _{n \rightarrow \infty} \cot \left(S_{n}(x)\right)=x \), for all \( x0 \)
(c) The equation \( S_{3}(x)=\frac{\pi}{4} \) has a root in \( (0, \infty) \)
(d) \( \tan \left(S_{n}(x)\right) \leq \frac{1}{2} \), for all \( n \geq 1 \) and \( x0 \)
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