<p>Evaluate \(\displaystyle\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n\sqrt{\frac{k}{n}}\)</p>
Step-by-Step Solution
Key Concept: Riemann sum: lim(1/n)\Sigmaf(k/n) = \int_0^1 f(x)dx. Here f(x) = \sqrt{x.}
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<p>Recognize as Riemann sum with $f(x)=\sqrt{x}$, $x_k=k/n$, $\Delta x=1/n$:</p>
<p>$$\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n\sqrt{\frac{k}{n}} = \int_0^1\sqrt{x}\,dx = \left[\frac{2}{3}x^{3/2}\right]_0^1 = \boxed{\frac{2}{3}}$$</p>
Correct Answer: A