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JEE Main2016MathematicsLimitsActual

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

Options

  1. A9 e 2
  2. B3 log 3 - 2
  3. C18 e 4
  4. D27 e 2

Correct answer

D. 27 e 2

Step-by-step solution

lim n → ∞ n + 1 n + 2 … . 3 n n 2 n 1 n Let y = n + 1 n + 2 … .     n + 2 n n 2 n 1 n y = n + 1 n · n + 2 n … … n + 2 n n 1 n log y = 1 n log 1 + 1 n + log 1 + 2 n + … . . log 1 + 2 n n = 1 n ∑ r = 1 2 n ln 1 + r / n Replace ∑ → ∫ & r n → x & 1 n → d x As n → ∞ log y =   ∫ 0 2 log 1 + x d x Integrating by parts, we get, log y = ∫ 0 2 1 · log 1 + x d x = x · log 1 + x 0 2 - ∫ 0 2

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