The limit L = lim n → ∞ ⁡ ∑ r = 1 n n n 2 + r 2 satisfies, π L > 6, π L 2, π L > 3, π L 1

Given limit is L = lim n → ∞ ∑ r = 1 n 1 1 + r n 2 ⋅ 1 n = ∫ 0 1 d x 1 + x 2 = tan - 1 ⁡ x 0 1 = π 4

The limit L = lim n → ∞ ⁡ ∑ r = 1 n n n 2 + r 2 satisfies

Question: Answered & Verified by Expert

The limit

L = \lim_{n \to \infty} \sum_{r = 1}^{n} \frac{n}{n^{2} + r^{2}}

satisfies

MathematicsDefinite IntegrationJEE Main

Options:

Solution:

1634 Upvotes Verified Answer

The correct answer is:

\frac{π}{L} > 3

Given limit is

L = \lim_{n \to \infty} \sum_{r = 1}^{n} \frac{1}{1 + {(\frac{r}{n})}^{2}} \cdot \frac{1}{n}

= \int_{0}^{1} \frac{d x}{1 + x^{2}} = {[\tan^{- 1} x]}_{0}^{1}

= \frac{π}{4}

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