For any positive integer \( n \), defined \( f_{n}:(0, \infty) \rightarrow R \) as
\( \mathrm{P} \) \( f_{n}(x)=\sum_{j=1}^{n} \tan ^{-1}\left(\frac{1}{1+(x+j)(x+j-1)}\right) \) for all
W \( x \in(0, \infty) \).
(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 is (are) true?
(A) \( \sum_{j=1}^{5} \tan ^{2}\left(f_{j}(0)\right)=55 \)
(B) \( \sum_{j=1}^{10}\left(1+f_{j}^{\prime}(0) \sec ^{2}\left(f_{j}(0)\right)\right)=10 \)
(C) For any fixed positive integer \( n \),
\[
\lim _{x \rightarrow \infty} \tan \left(f_{n}(x)\right)=\frac{1}{n}
\]
(D) For any fixed positive integer \( n \),
\[
\lim _{x \rightarrow \infty} \sec ^{2}\left(f_{n}(x)\right)=1
\]
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