lim n → ∞ 1 + 2 + … + n n 3 2 =

The expression is given as,

$I=\underset{n\to \infty }{\lim }\frac{\sqrt{1}+\sqrt{2}+\dots .+\sqrt{n}}{n^{{\displaystyle \frac{3}{2}}}}$

$=\underset{n\to \infty }{\lim }\frac{\sqrt{1}+\sqrt{2}+\dots ..+\sqrt{n}}{n\sqrt{n}}$

$=\underset{n\to \infty }{\lim }\frac{1}{n}\left[\sqrt{\frac{1}{n}}+\sqrt{\frac{2}{n}}+\dots +\sqrt{\frac{n}{n}}\right]$

$=\sum _{r=1}^n\underset{n\to \infty }{\lim }\frac{1}{n}\sqrt{\frac{1}{n}}$

$={\int }_0^1\sqrt{x}dx$

$=\frac{2}{3}{\left[x^{\frac{3}{2}}\right]}_0^1=\frac{2}{3}$