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JEE Main202125 Feb 2021Evening ShiftMathematicsDefinite IntegrationActual

lim n → ∞ 1 n + n n + 1 2 + n n + 2 2 + … + n 2 n - 1 2 is equal to

Options

  1. A1 2
  2. B1 4
  3. C1 3
  4. D1

Correct answer

A. 1 2

Step-by-step solution

lim n → ∞ 1 n + n n + 1 2 + n n + 2 2 + … + n 2 n - 1 2 = lim n → ∞ ∑ r = 0 n - 1 n n + r 2 = lim n → ∞ ∑ r = 0 n - 1 n n 2 + 2 n r + r 2 = lim n → ∞ 1 n ∑ r = 0 n - 1 1 r / n 2 + 2 r / n + 1 = ∫ 0 1 d x x + 1 2 = - 1 x + 1 0 1 = 1 2

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