Question

$\lim_{n \to \infty} \left[ \frac{1}{n+1} + \frac{1}{n+2} + \frac{1}{n+3} + \ldots + \frac{1}{n+6n} \right]$ is

• A. $\log 2$
• B. $\log(1+\sqrt{5})$
• C. $\log 6$
• D. $0$

Hint:

Riemann sum: $\lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} f(k/n) = \int_0^1 f(x)dx$.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This is a limit of a sum that can be expressed as a Riemann sum.
Step 2: Detailed Explanation:
$\lim_{n \to \infty} \sum_{k=1}^{5n} \frac{1}{n+k} = \lim_{n \to \infty} \frac{1}{n} \sum_{k=1}^{5n} \frac{1}{1 + k/n}$.
This is $\int_0^5 \frac{dx}{1+x} = [\log(1+x)]_0^5 = \log 6 - \log 1 = \log 6$.
Step 3: Final Answer:
The limit is $\log 6$.