$\lim _{n \rightarrow \infty}\left(\frac{\sqrt{1}+2 \sqrt{2}+3 \sqrt{3}+\ldots+n \sqrt{n}}{n^{\frac{5}{2}}}\right)=$

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Aicharya · Accepted Answer

We have, $ \lim _{x ightarrow \infty}\left[\frac{\sqrt{1}+2 \sqrt{2}+3 \sqrt{3}+\ldots+n \sqrt{n}}{n^{5 / 2}}ight] $ $\begin{aligned} & =\lim _{x ightarrow \infty} \sum_{r=1}^n\left(\frac{r}{n}ight)^{3 / 2} \cdot \frac{1}{n} \ & =\int_0^1 x^{3 / 2} d x=\left[\frac{x^{5 / 2}}{5 / 2}ight]_0^1=\frac{2}{5}\end{aligned}$

$\lim _{n \rightarrow \infty}\left(\frac{\sqrt{1}+2 \sqrt{2}+3 \sqrt{3}+\ldots+n…

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