Q116 / 160 · AP EAMCET 2018 (22 Apr Shift 1) · Mathematics

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

  • A
    1
  • B
    $\frac{5}{2}$
  • C
    0
  • Correct answer
    D
    $\frac{2}{5}$

Solution

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

Hence, the correct answer is option: 'D'.

Relevant Questions