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JEE Advanced2018MathematicsLimitsActual

For each positive integer n , let y n = 1 n ( ( n + 1 ) ( n + 2 ) ... ( n + n ) ) 1 n . If lim n → ∞ ⁡ y n = L , then the value of L (where x is the greatest integer less than or equal to x ) is ____

Correct answer

0

Step-by-step solution

y n = 1 + 1 n 1 + 2 n … . 1 + n n 1 n y n = ∏ r = 1 n 1 + r n 1 n ln y n = 1 n ∑ r = 1 n ln ⁡ 1 + r n ⇒     lim n → ∞ ⁡ ln ⁡ ( y n ) = lim n → ∞ ⁡ ∑ r = 1 n 1 n ln ⁡ 1 + r n ⇒   ln L = ∫ 0 1 ln ⁡ 1 + x d x ⇒   ln L = ln 4 e ⇒     L = 4 e L = 1

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