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BITSAT2018MathematicsSequences and SeriesActual

If ∑ r = 1 n t r = n n + 1 n + 2 n + 3 8 , where t r denotes the r t h term of a series, then lim n → ∞ ∑ r = 1 n 1 t r is

Options

  1. A1 8
  2. B1 4
  3. C1 2
  4. D1

Correct answer

C. 1 2

Step-by-step solution

Given, ∑ r = 1 n t r = n n + 1 n + 2 n + 3 8 = S n (say) ∴   ∑ r = 1 n - 1 t r = n - 1 n n + 1 n + 2 8 = S n - 1 Now, t n = S n - S n - 1 = n n + 1 n + 2 2 ∴   lim n → ∞ ∑ r = 1 n 1 t r = lim n → ∞ ∑ r = 1 n 2 n n + 1 n + 2 = lim n → ∞ ∑ r = 1 n 1 n n + 1 - 1 n + 1 n + 2 = - lim n → ∞ ∑ r = 1 n 1 n + 1 n + 2 - 1 n n + 1 = - lim n → ∞ 1 n + 1 n + 2 - 1 2 = - 0 - 1 2 = 1 2

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