SATHEE: Indefinite Integration 4 Question 3

Indefinite Integration 4 Question 3

3. $lim_{n \to \infty} \frac{(n + 1) (n + 2) \dots 3 n}{n^{2 n}}^{1 / n}$ is equal to

(a) $\frac{18}{e^{4}}$

(b) $\frac{27}{e^{2}}$

(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 $l = lim_{n \to \infty} \frac{(n + 1) \cdot (n + 2) \dots (3 n)}{n^{2 n}} \frac{1}{n}$

$\begin{aligned} = lim_{n \to \infty} \frac{(n + 1) \cdot (n + 2) \dots (n + 2 n)^{\frac{1}{n}}}{n^{2 n}} \\ = lim_{n \to \infty} \frac{n + 1}{n} \frac{n + 2}{n} \dots \frac{n + 2 n}{n} \end{aligned}$

Taking log on both sides, we get

$\begin{aligned} \log l = lim_{n \to \infty} \frac{1}{n} \log 1 + \frac{1}{n} 1 + \frac{2}{n} \dots 1 + \frac{2 n}{n} \\ \Rightarrow \log l = lim_{n \to \infty} \frac{1}{n} \\ \log 1 + \frac{1}{n} + \log 1 + \frac{2}{n} + \dots + \log 1 + \frac{2 n}{n} \\ \Rightarrow \log l = lim_{n \to \infty} \frac{1}{n} \sum_{r = 1}^{2 n} \log 1 + \frac{r}{n} \\ \Rightarrow \log l = \int_{0}^{2} \log (1 + x) d x \\ \Rightarrow \log l = \log (1 + x) \cdot x - \int \frac{1}{1 + x} \cdot x d x \\ \Rightarrow \log l = [\log (1 + x) \cdot x]_{0}^{2} - \int_{0}^{2} \frac{x + 1 - 1}{1 + x} d x \\ \Rightarrow \log l = 2 \cdot \log 3 - \int_{0}^{2} 1 - \frac{1}{1 + x} d x \\ \Rightarrow \log l = 2 \cdot \log 3 - [x - \log | 1 + x |]_{0}^{2} \\ \Rightarrow \log l = 2 \cdot \log 3 - [2 - \log 3] \\ \Rightarrow \log l = 3 \cdot \log 3 - 2 \\ \Rightarrow \log l = \log 27 - 2 \\ ∴ l = e^{\log 27 - 2} = 27 \cdot e^{- 2} = \frac{27}{e^{2}} \end{aligned}$

Indefinite Integration 4 Question 3

3. $lim_{n \to \infty} \frac{(n + 1) (n + 2) \dots 3 n}{n^{2 n}}^{1 / n}$ is equal to

Objective Questions II

Answer:

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Indefinite Integration 4 Question 3

3. limn→∞(n+1)(n+2)…3nn2n1/n is equal to

Objective Questions II

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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3. $lim_{n \to \infty} \frac{(n + 1) (n + 2) \dots 3 n}{n^{2 n}}^{1 / n}$ is equal to