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EEJ MAIN Mathematics QUESTION #6971
Question 1

Evaluate $\displaystyle\lim_{n\to\infty}\left(\dfrac{(n+1)(n+2)\cdots3n}{n^{2n}}\right)^{1/n}$

  • $\dfrac{27}{e^2}$✔️
  • $\dfrac{9}{e^2}$
  • $3\ln3-2$
  • $\dfrac{18}{e^4}$
Correct Answer Explanation

$\ln L = \lim_{n\to\infty}\dfrac{1}{n}\sum_{r=1}^{2n}\ln\!\left(1+\dfrac{r}{n}\right) = \int_0^2\ln(1+x)\,dx$

$=\left[(1+x)\ln(1+x)-(1+x)\right]_0^2 = (3\ln3-3)-(-1) = 3\ln3-2$

$L = e^{3\ln3-2} = \dfrac{27}{e^2}$