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JEE Main20231 Feb 2023Morning ShiftMathematicsDefinite IntegrationActual

lim n → ∞ 1 1 + n + 1 2 + n + 1 3 + n + … + 1 2 n is equal to :-

Options

  1. A0
  2. Blog e 2
  3. Clog e 3 2
  4. Dlog e 2 3

Correct answer

B. log e 2

Step-by-step solution

We have, lim n → ∞ 1 1 + n + 1 2 + n + 1 3 + n + … + 1 2 n = lim n → ∞ 1 1 + n + 2 1 + n + 3 1 + n + … . + 1 n + n = lim n → ∞ ∑   r = 1 n 1 n + r = lim n → ∞   ∑ r = 1 n 1 n 1 1 + r n = ∫ 0 1 1 1 + x d x = log e 1 + x 0 1 = log e 2

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