Definite Integration
Standard Limit
Grade 12
Question:
<p>Evaluate \(\displaystyle\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n\frac{1}{1+k/n}\) [JEE Main 2016]</p>
\(\ln 2\)
\(1\)
\(2\ln 2\)
\(\ln 3\)
Step-by-Step Solution
Key Concept: Riemann sum \to \int_0^1 1/(1+x) dx = [ln(1+x)]_0^1 = ln 2.
<div class='solution'>
<p>$$\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n\frac{1}{1+k/n}=\int_0^1\frac{dx}{1+x}=[\ln(1+x)]_0^1=\ln 2$$</p>
Correct Answer: A