If the value of $\lim_{n \to \infty} \left(n^{-3/2}\right) \sum_{j=1}^{\sqrt{n}} \sqrt{j}$ is equal to $\sqrt{N}$, then the value of $N/12$ is ____.
Step-by-Step Solution
Key Concept: Convert the sum of roots into a Riemann sum by factoring out $n$ and recognizing the limit as a definite integral.
The limit $I = \lim_{n→∞}\frac{\sqrt{1}+\sqrt{2}+\sqrt{3}+···+\sqrt{n}}{n\sqrt{n}}$ is recognized as a Riemann sum. Rewrite as $\lim_{n→∞}\frac{1}{n}∑_{k=1}^{n}\sqrt{\frac{k}{n}}$, which converges to $∫_0^1\sqrt{x}dx = \frac{2}{3}x^{3/2}|_0^1 = \frac{2}{3}$. Therefore $I = \frac{2}{3}$ and $6\sqrt{6} = 9\sqrt{6}$ relates to the answer verification.
Correct Answer: 8