For any positive integer n , define f n : 0 , ∞ → R as f n x = ∑ i = 1 n tan - 1 1 1 + x + j x + j - 1 for all x ∈ 0 , ∞ . ( Here, the inverse trigonometric function tan - 1 ⁡ x assumes values in - π 2 , π 2 ) Then, which of the following statement(s) is (are) TRUE?

Step-by-step solution

f n x = ∑ j = 1 n tan - 1 x + j - x + j - 1 1 + x + j x + j - 1 f n x = ∑ j = 1 n tan - 1 x + j - tan - 1 x + j - 1 f n x = tan - 1 x + n - tan - 1 x ∴ tan f n x = tan tan - 1 x + n - tan - 1 x tan f n x = tan tan - 1 x + n - x 1 + x x + n tan f n x = n 1 + x 2 + n x , so that lim x → ∞ tan f n x = 0 ∴ sec 2 f n x = 1 + tan 2 f n x sec 2 f n x = 1 + n 1 + x 2 + n x 2 lim x → ∞ sec 2 f n x = lim x → ∞ 1 + n 1 + x 2 + n x 2 = 1 Note that since 0 is not in do