For any positive integer $n$, define $f_n:\left(0, \infty \right)\to R$ as

$f_n\left(x\right)={\sum }_{i=1}^n{\tan }^{-1}\left(\frac{1}{1+\left(x+j\right)\left(x+j-1\right)}\right)$ for all $x\in \left(0,\infty \right)$ .

$($Here, the inverse trigonometric function ${\tan }^{-1}x$ assumes values in $\left(-\frac{\pi }{2}, \frac{\pi }{2}\right)$$)$

Then, which of the following statement(s) is (are) TRUE?

• A

  ${\sum }_{j=1}^5{\tan }^2\left(f_j\left(0\right)\right)=55$
• B

  ${\sum }_{j=1}^{10}\left(1+{f_j}^{\text{'}}\left(0\right)\right){\sec }^2\left(f_j\left(0\right)\right)=10$
• C

  For any fixed positive integer $n$,$\underset{x\to \infty }{\lim }\tan \left(f_n\left(x\right)\right)=\frac{1}{n}$
• Correct answer
  D

  For any fixed positive integer $n$, $\underset{x\to \infty }{\lim }{\sec }^2\left(f_n\left(x\right)\right)=1$

Hence, the correct answer is option: 'D'.