Definite Integration
Limit as Riemann Sum
Grade 12
Question:
<p>The value of: \(\lim_{n \to \infty} \left( \frac{1}{n\sqrt{n+1}} + \frac{1}{n\sqrt{n+2}} + \frac{1}{n\sqrt{n+3}} + \ldots + \frac{1}{n\sqrt{2n}} \right)\) is:</p>
<p>(a) \(\sqrt{2} - 1\)</p>
<p>(b) \(2(\sqrt{2} - 1)\)</p>
<p>(c) \(\sqrt{2} + 1\)</p>
<p>(d) \(2(\sqrt{2} + 1)\)</p>
Step-by-Step Solution
Key Concept: Convert the limit of a sum into a definite integral using Riemann sum interpretation where the sum represents $\frac{1}{n}$ times terms evaluated at points between 1 and 2.
<p>This limit can be evaluated by recognizing it as a Riemann sum. As $n \to \infty$, the sum approximates a definite integral.</p>
Correct Answer: B