<s>[INST] Finding $\lim_{n\rightarrow \infty} \frac{\sqrt{1}+\sqrt{2}+\sqrt{3}+\cdots+\sqrt{n}}{n^{3/2}}$ $$\lim_{n\rightarrow \infty} \frac{\sqrt{1}+\sqrt{2}+\sqrt{3}+\cdots+\sqrt{n}}{n^{3/2}}$$
How to find the limit? Any help will be appreciated.
 [/INST] By Stolz we have:
$$\lim_{n\rightarrow \infty} \frac{\sqrt{1}+\sqrt{2}+\sqrt{3}+\cdots+\sqrt{n}}{n^{3/2}}=\lim_{n\rightarrow \infty} \frac{\sqrt{n}}{n^{3/2}-(n-1)^{\frac{3}{2}}}=\lim_{n\rightarrow \infty} \frac{\sqrt{n}\left(n^{3/2}+(n-1)^{\frac{3}{2}}\right)}{n^3-(n-1)^3}=$$
$$=\lim_{n\rightarrow\infty}\frac{n^2+\sqrt{n(n-1)^3}}{3n^2-3n+1}=\frac{2}{3}$$
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