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?