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JEE Main202127 Aug 2021Morning ShiftMathematicsDefinite IntegrationActual

If U n = 1 + 1 n 2 1 + 2 2 n 2 2 … 1 + n 2 n 2 n , then lim n → ∞ U n - 4 n 2 is equal to

Options

  1. A16 e 2
  2. B4 e
  3. C4 e 2
  4. De 2 16

Correct answer

D. e 2 16

Step-by-step solution

U n = ∏ r = 1 n 1 + r 2 n 2 r L = lim n → ∞ U n - 4 n 2 log L = lim n → ∞ - 4 n 2 ∑ r = 1 n   log 1 + r 2 n 2 r log   L = lim n → ∞   ∑ r = 1 n   - 4 r n · 1 n   log 1 + r 2 n 2 log   L = - 4 ∫ 0 1 x   log 1 + x 2 d x Put 1 + x 2 = t 2 x   d x = d t = - 2 ∫ 1 2 log   t d t = - 2 t   log   t - t 1 2 ⇒ log   L = - 2 2 log   2 - 1 ∴   L = e - 2 2 log   2 - 1 = e

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