Duration: 0:00

15 views

0

For any positive integer \( n \), define \( f_{n}:(0, \infty) \rightarrow R \) as
\( \mathrm{P} \) \( f_{n}(x)=\sum_{j=1}^{n} \tan ^{-1}\left(\frac{1}{1+(x+j)(x+j-1)}\right) \) for all \( x \in(0, \infty) \).
(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(s) is (are) TRUE ?
(A) \( \sum_{j=1}^{5} \tan ^{2}\left(f_{j}(0)\right)=55 \)
(B) \( \sum_{j=1}^{10}\left(1+f_{j}^{\prime}(0)\right) \sec ^{2}\left(f_{j}(0)\right)=10 \)
(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 \( n \), \( \lim _{x \rightarrow \infty} \sec ^{2}\left(f_{n}(x)\right)=1 \)
📲 PW App Link - https://bit.ly/YTAI_PWAP
🌐 PW Website - https://www.pw.live/