BITSAT2018MathematicsInverse Trigonometric FunctionsActual
The value of S = ∑ n = 1 ∞ tan - 1 2 n n 4 + n 2 + 2 is equal to
Options
- Aπ 2
- Bπ
- Cπ 4
- DNone of these
Correct answer
C. π 4
Step-by-step solution
Given, S = ∑ n = 1 ∞ tan - 1 2 n n 4 + n 2 + 2       . . . i Let n 4 + n 2 + 1 = n 2 2 + 1 2 + 2 n 2 1 - n 2 = n 2 + 1 2 - n 2 = n 2 + n + 1 n 2 - n + 1       . . . ii Let u n = tan - 1 2 n n 4 + n 2 + 2 = tan - 1 2 n 1 + n 4 + n 2 + 1 = tan - 1 n 2 + n + 1 - n 2 - n + 1 1 + n 2 + n + 1 n 2 - n + 1 u n = tan - 1 n 2 + n + 1 - tan - 1 n 2 - n + 1       . . . iii On putting n = 1 ,   2 ,   3 , … … successively in Eq. (iii), we get u 1 = ta