Question

For any positive integer $n$, let $S_{n}:(0, \infty) \rightarrow R$ be defined by
$S_{n}(x)=\displaystyle\sum_{k=1}^{n} \cot ^{-1}\left(\frac{1+k(k+1) x^{2}}{x}\right),$
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. $\displaystyle\lim _{n \rightarrow \infty} \cot \left(S_{n}(x)\right)=x$, for all $x>0$
• 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 $x>0$

Answer 1

Answer: (A), (B)

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

Answer 2

(A) $S _{10}( x )=\frac{\pi}{2}-\tan ^{-1}\left(\frac{1+11 x ^{2}}{10 x }\right)$, for all $x >0$
(B) $\displaystyle\lim _{n \rightarrow \infty} \cot \left(S_{n}(x)\right)=x$, for all $x>0$