Indefinite Integration 4 Question 3
3. $\lim _{n \rightarrow \infty} \frac{(n+1)(n+2) \ldots 3 n}{n^{2 n}}{ }^{1 / n}$ is equal to
(a) $\frac{18}{e^{4}}$
(b) $\frac{27}{e^{2}}$
(c) $\frac{9}{e^{2}}$
(d) $3 \log 3-2$
Objective Questions II
(One or more than one correct option)
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Answer:
Correct Answer: 3. (b)
Solution:
- Let $l=\lim _{n \rightarrow \infty} \frac{(n+1) \cdot(n+2) \ldots(3 n)}{n^{2 n}} \frac{1}{n}$
$$ \begin{aligned} & =\lim _{n \rightarrow \infty} \frac{(n+1) \cdot(n+2) \ldots(n+2 n)^{\frac{1}{n}}}{n^{2 n}} \\ & =\lim _{n \rightarrow \infty} \frac{n+1}{n} \quad \frac{n+2}{n} \ldots \frac{n+2 n}{n} \end{aligned} $$
Taking log on both sides, we get
$$ \begin{aligned} & \log l=\lim _{n \rightarrow \infty} \frac{1}{n} \log 1+\frac{1}{n} 1+\frac{2}{n} \ldots 1+\frac{2 n}{n} \\ & \Rightarrow \quad \log l=\lim _{n \rightarrow \infty} \frac{1}{n} \\ & \log 1+\frac{1}{n}+\log 1+\frac{2}{n}+\ldots+\log 1+\frac{2 n}{n} \\ & \Rightarrow \quad \log l=\lim _{n \rightarrow \infty} \frac{1}{n} \sum _{r=1}^{2 n} \log 1+\frac{r}{n} \\ & \Rightarrow \quad \log l=\int _0^{2} \log (1+x) d x \\ & \Rightarrow \quad \log l=\log (1+x) \cdot x-\int \frac{1}{1+x} \cdot x d x \\ & \Rightarrow \quad \log l=[\log (1+x) \cdot x] _0^{2}-\int _0^{2} \frac{x+1-1}{1+x} d x \\ & \Rightarrow \quad \log l=2 \cdot \log 3-\int _0^{2} 1-\frac{1}{1+x} d x \\ & \Rightarrow \quad \log l=2 \cdot \log 3-[x-\log |1+x|] _0^{2} \\ & \Rightarrow \quad \log l=2 \cdot \log 3-[2-\log 3] \\ & \Rightarrow \quad \log l=3 \cdot \log 3-2 \\ & \Rightarrow \quad \log l=\log 27-2 \\ & \therefore \quad l=e^{\log 27-2}=27 \cdot e^{-2}=\frac{27}{e^{2}} \end{aligned} $$
