calculus - Find $lim _{ nrightarrow infty }{ sum _{ k=1 }^{ n }{ frac { sqrt { k } }{ { n }^{ frac { 3 }{ 2 } } } } }$

Need help find the limit of $\lim _{ n\rightarrow \infty }{ \sum _{ k=1 }^{ n }{ \frac { \sqrt { k } }{ { n }^{ \frac { 3 }{ 2 } } } } }$

Now my intuition is that using Stolz-Cesaro

$\lim _{ n\rightarrow \infty }{ \sum _{ k=1 }^{ n }{ \frac { \sqrt { k } }{ { n }^{ \frac { 3 }{ 2 } } } } }=\lim _{ n\rightarrow \infty }{ \frac { 1 }{ n } \sum _{ k=1 }^{ n }{ \sqrt { \frac { k }{ n } } } =1 }$

Is it correct or not?

Answer

how about using Riemann sums??

$$\lim_{n\to\infty} \sum_{k=1}^n \frac{\sqrt k}{n^{3/2}} = \lim_{n\to\infty}\frac 1 n \sum_{k=1}^n \sqrt{\frac k n} = \int_0^1 \sqrt x dx$$