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NTA Abhyas JEE Main2020MathematicsLimitsPractice

If ∑ r = 1 n t r = 1 6 n n + 1 n + 2 ,   ∀ n ≥ 1 , then the value of l i m n → ∞ ∑ r = 1 n 1 t r is equal to

Options

  1. A2
  2. B3
  3. C3 2
  4. D6

Correct answer

A. 2

Step-by-step solution

We have, for n ≥ 1 ,   t n = ∑ r = 1 n t r - ∑ r = 1 n - 1 t r = 1 6 n n + 1 n + 2 - 1 6 n - 1 n n + 1 = 1 6 n n + 1 n + 2 - n + 1 = 1 2 n n + 1 . ⇒ 1 t r = 2 r r + 1 = 2 1 r - 1 r + 1 ⇒ ∑ r = 1 n 1 t r = 2 1 - 1 n + 1 ⇒ l i m n → ∞ ∑ r = 1 n 1 t r = 2

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