For any positive integer n, define f_n:(0,\infty)\to\mathbb{R} as f_n(x)=\sum^{n}_{j=1}\tan^{-1}(1/{1+(x+j)(x+j-1)}) for all x∊(0,\infty)(Here, the inverse trigonometric function \tan^{-1}x assume values in (-\pi/2,\pi/2)) Then, which of the following statement(s) is (are) TRUE? [JEE Adv 2018 P2]

Correct Answer is : For any fixed positive integer n, limx→∞sec2(fn(x))=1

\sum^{10}_{j=1}(1+f'_1(0))\sec^2(f_j(0))=10

For any fixed positive integer n, \text{lim}_{x\to\infty}\tan(f_n(x))=1/n

For any fixed positive integer n, \text{lim}_{x\to\infty}\sec^2(f_n(x))=1

Inverse Trigonometric Functions

Section: Mathematics

Question : 7 of 14

Marks: +1, -0

For any positive integer n, define fn:(0,∞)→ℝ as

fn(x)=

∑

j=1

tan−1(

1+(x+j)(x+j−1)

) for all x∊(0,∞)

(Here, the inverse trigonometric function tan−1x assume values in (−

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

[JEE Adv 2018 P2]

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