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

Question

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, 2, 3, 3 2, 6

MARKS App · Accepted Answer

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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