ExamHabit
Practise free

JEE Main 2016

JEE Main 2016 Mathematics question on Definite Integrals and Areas

lim⁡n→∞((n+1)(n+2)⋯3nn2n)1/n\lim_{n \to \infty} \left( \frac{(n+1)(n+2)\cdots 3n}{n^{2n}} \right)^{1/n} is equal to
  1. A.
    18e4\frac{18}{e^4}
  2. B.
    27e2\frac{27}{e^2}
  3. C.
    9e2\frac{9}{e^2}
  4. D.
    3log⁡3−23\log 3 - 2
Show answer and solution

Correct option: B

Solution

Take the log of the limit. Let L = lim of (1/n)·ln[∏(k=1 to 2n) (n+k) / n^(2n)].

ln expression = (1/n)·[Σ(k=1 to 2n) ln(n+k) - 2n·ln n]
= (1/n)·Σ(k=1 to 2n) ln((n+k)/n)
= (1/n)·Σ(k=1 to 2n) ln(1 + k/n)

This is a Riemann sum for ∫₀² ln(1+x)dx with step 1/n.

∫₀² ln(1+x)dx = [(1+x)ln(1+x) - (1+x)]₀² = [3ln3 - 3] - [0 - 1] = 3ln3 - 2.

So ln L = 3ln3 - 2 = ln(27) - 2, giving L = 27/e².

[JEE Main 2016]

Practise Definite Integrals and Areas the way it is asked

A timed set of real JEE Main Definite Integrals and Areas questions, graded as the exam grades. No sign-up.

Start a Definite Integrals and Areas test

More: all JEE Main Definite Integrals and Areas questions · JEE Main 2016 · Mathematics chapter list