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AP EAMCET201825 Apr 2018Morning ShiftMathematicsDefinite IntegrationActual

If a and b are positive integers such that b > a , then lim n → ∞ 1 n a + 1 n a + 1 + 1 n a + 2 + … + 1 n b =

Options

  1. Alog b a
  2. Blog a b
  3. Clog ( a b )
  4. Dlog ( a + b )

Correct answer

A. log b a

Step-by-step solution

We have, lim n → ∞ 1 n a + 1 n a + 1 + 1 n a + 2 + … + 1 n b = lim n → ∞ 1 n a + 1 n a + 1 + 1 n a + 2 + … + 1 n a + n b - a = lim n → ∞ 1 n a + 1 n a + 1 + 1 n a + 2 + … + 1 n a + n b - a = lim n → ∞ 1 n 1 a + 1 a + 1 n + 1 a + 2 n + … + 1 a + n b - a n = lim n → ∞ ∑ r = 0 n b - a 1 n 1 a + r n Now, replacing 1 n → d x ,   r n → x and converting limits, we get = ∫ 0 b - a d x a + x = log a + x 0 b - a =

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