Mathematics - Definite Integration Question with Solution | TestHub

$$\underset{n\to \infty }{\lim }\frac{{\sum }_{r=1}^n\sqrt{r}{\sum }_{r=1}^n\frac{1}{\sqrt{r}}}{\frac{n(n+1)}{2}}=\underset{n\to \infty }{\lim }\frac{{\sum }_{r=1}^n\sqrt{\frac{r}{n}}{\sum }_{r=1}^n\frac{1}{\sqrt{\frac{r}{n}}}}{\frac{n^2}{2}\left(1+\frac{1}{n}\right)}$$

$$\begin{array}{rl}& =2\times {\int }_0^1\sqrt{x}dx{\int }_0^1\frac{1}{\sqrt{x}}dx\\ & =2\times {\left[\frac{x^{3/2}}{\frac{3}{2}}\right]}_0^1{\left[\frac{x^{1/2}}{\frac{1}{2}}\right]}_0^1\\ & =2\times \frac{2}{3}\times 2=\frac{8}{3}\\ & \therefore k=8\end{array}$$