Problem on limits $\lim_{n \to \infty} \frac{\sqrt{1}+\sqrt{2}+\sqrt{3}+\sqrt{4}+...+\sqrt{n}}{n\sqrt{n}}$

Question

I have the problem with the following limit:

$$\lim_{n \to ∞} \frac{\sqrt{1}+\sqrt{2}+\sqrt{3}+\sqrt{4}+...+\sqrt{n}}{n\sqrt{n}}$$

Answer 1

This is a Riemann sum: $$ \lim_{n\to\infty} \frac{1}{n}\sum_{k=1}^n \sqrt{\frac{k}{n}} = \int_0^1\sqrt{x}dx = \frac{2}{3}. $$