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JEE Main201912 Jan 2019Evening ShiftMathematicsDefinite IntegrationActual

lim n → ∞ n n 2 + 1 2 + n n 2 + 2 2 + n n 2 + 3 2 + . . … . + 1 5 n 2 is equal to

Options

  1. Aπ 4
  2. Btan - 1 ⁡ 2
  3. Cπ 2
  4. Dtan - 1 ⁡ ( 3 )

Correct answer

B. tan - 1 ⁡ 2

Step-by-step solution

To find, lim n → ∞ n n 2 + 1 2 + n n 2 + 2 2 + … + n n 2 + 2 n 2 = ∑ r = 1 2 n n n 2 + r 2 = 1 n ∑ r = 1 2 n 1 1 + r n 2     = ∫ 0 2 1 1 + x 2 d x = tan - 1 ⁡ x 0 2 = tan - 1 ⁡ 2 .

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