JEE Advanced2021MathematicsInverse Trigonometric FunctionsActual
For any positive integer n , let S n : ( 0 , ∞ ) → R be defined by S n x = ∑ k = 1 n cot - 1 1 + k ( k + 1 ) x 2 x where for any x ∈ R , cot - 1 ( x ) ∈ ( 0 , π ) and tan - 1 ( x ) ∈ - π 2 , π 2 . Then which of the following statements is (are) TRUE ?
Options
- AS 10 x = π 2 - tan - 1 1 + 11 x 2 10 x , for all x > 0
- Blim n → ∞ cot S n ( x ) = x , for all x > 0
- CThe equation S 3 ( x ) = π 4 has a root in ( 0 , ∞ )
- Dtan S n ( x ) ≤ 1 2 , for all n ≥ 1 and x > 0
Correct answer
A. S 10 x = π 2 - tan - 1 1 + 11 x 2 10 x , for all x > 0
Step-by-step solution
S n x = ∑ k = 1 n cot - 1 1 + k ( k + 1 ) x 2 x = ∑ k = 1 n cot - 1 1 + k x ( k + 1 ) x ( k + 1 ) x - k x = ∑ k = 1 n tan - 1 ( k + 1 ) x - k x 1 + k x ( k + 1 ) x = ∑ k = 1 n tan - 1 ( k + 1 ) x - tan - 1 k x = tan - 1 ( n + 1 ) x - tan - 1 n x + … + tan - 1 3 x - tan - 1 2 x + tan - 1 2 x - tan - 1 x = tan - 1 ( n + 1 ) x - tan - 1 x = tan - 1 ( n + 1 ) x - x 1 + ( n + 1 ) x 2 = tan - 1 n x 1 + ( n + 1 ) x 2 1. S 10 x = tan - 1 10 x 1 + 11 x 2 = π 2 - cot - 1 10 x 1 + 11 x 2 = &