For any positive integer $n$, define $f_n:(0, \infty) \rightarrow R$ as
$ f_n(x)=\sum_{j=1}^n \tan ^{-1}\left(\frac{1}{1+(x+j)(x+j-1)}\right) \text { for all } x \in(0, \infty) $
(Here, the inverse trigonometric function $\tan ^{-1} \mathrm{x}$ assumes values in $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ )
Then, which of the following statement(s) is (are) TRUE ?