JEE Main202226 Jun 2022Evening ShiftMathematicsSequences and SeriesActual
If A = ∑ n = 1 ∞ 1 3 + - 1 n n and B = ∑ n = 1 ∞ - 1 n 3 + - 1 n n , then A B is equal to
Options
- A11 9
- B1
- C- 11 9
- D- 11 3
Correct answer
C. - 11 9
Step-by-step solution
Given, A = ∑ n = 1 ∞ 1 3 + - 1 n n = 1 2 + 1 4 2 + 1 2 3 + 1 4 4 + ⋯ = 1 2 + 1 2 3 + ⋯ + 1 4 2 + 1 4 4 + ⋯ = 1 2 1 - 1 4 + 1 16 1 - 1 16 = 2 3 + 1 15 = 11 15 B = ∑ n = 1 ∞ - 1 n 3 + - 1 n n = - 1 2 + 1 4 2 - 1 2 3 + 1 4 4 - ⋯ = - 1 2 + 1 2 3 + ⋯ + 1 4 2 + 1 4 4 + ⋯ = - 1 2 1 - 1 4 + 1 16 1 - 1 16 = - 2 3 + 1 15 = - 9 15 Hence, A B = - 11 9