A) 1
B) 4
C) 6
D) 8
Correct Answer: D
Solution :
\[\underset{n\to \infty }{\mathop{\lim }}\,\frac{\sum\limits_{r=1}^{n}{\sqrt{r}}\sum\limits_{r=1}^{n}{\frac{1}{\sqrt{r}}}}{\frac{n(n+1)}{2}}\underset{n\to ki}{\mathop{\lim }}\,\frac{\sum\limits_{r=1}^{n}{\sqrt{\frac{r}{n}}}\sum\limits_{r=1}^{n}{\frac{1}{\sqrt{\frac{r}{n}}}}}{\frac{{{n}^{2}}}{2}\left( 1+\frac{1}{n} \right)}\] \[=2\times \int\limits_{0}^{1}{\sqrt{x}}dx\int\limits_{0}^{1}{\frac{1}{\sqrt{x}}}dx\] \[=2\times \left[ \frac{{{x}^{3/2}}}{\frac{3}{2}} \right]_{0}^{1}\left[ \frac{{{x}^{1/2}}}{\frac{1}{2}} \right]_{0}^{1}=2\times \frac{2}{3}\times 2=\frac{8}{3}\] \[\therefore \]\[k=8\]
You need to login to perform this action.
You will be redirected in
3 sec
