For any positive integer n, define fn : (0, \infty) → R as \left(\text{f}\right)_{\text{n}} \left(\right. \text{x} \left.\right) \textrm{ } = \sum_{j = 1}^{n} \left(tan\right)^{- 1} \left(\right. \frac{1}{1 + \left(\right. x + j \left.\right) \textrm{ } \left(\right. x + j - 1 \left.\right)} \left.\right)for all x\in \left(\right. 0 , \infty \left.\right) (Here, the inverse trigonometric function tan–1 x assumes values in .) Then, which of the following statement(s) is(are) TRUE ?

Engineering

Mathematics

Properties of Inverse Trigonometric Function

Question

For any positive integer n, define f_n : (0, $\infty$ ) → R as $f_{n} (x) = \sum_{j = 1}^{n} \tan^{- 1} (\frac{1}{1 + (x + j) (x + j - 1)})$ for all x $\in (0, \infty)$

(Here, the inverse trigonometric function tan–1 x assumes values in .)

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

For any fixed positive integer n, $\underset{x \to \infty}{Lim} \tan (f_{n} (x)) = \frac{1}{n}$

$\sum_{j = 1}^{5} \tan^{2} (f_{j} (0)) = 55$

For any fixed positive integer n, $\underset{x \to \infty}{Lim} \sec^{2} (f_{n} (x)) = 1$

$\sum_{j = 1}^{10} (1 + f_{j}^{'} (0)) \sec^{2} (f_{j} (0)) = 10$

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$f_{n} (x) = \sum_{j = 1}^{n} \tan^{- 1} ((x + j)) - \tan^{- 1} (x + j - 1) = {tan}^{- 1} (x + n) - {tan}^{- 1} (x) = \tan^{- 1} (\frac{n}{1 + x^{2} + nx})$

(A) x = 0 is not in domain

(B) x = 0 is not in domain

(D) $\underset{x \to \infty}{Lim} 1 + {(\frac{n}{1 + x^{2} + nx})}^{2} = 1$