Mathematics - Limits Question with Solution | TestHub

Let$S=\underset{n\to \infty }{\lim }\left(\frac{1+2^{\frac{1}{3}}+3^{\frac{1}{3}}+\dots +n^{\frac{1}{3}}}{n^{\frac{7}{3}}\left(\frac{1}{{\left(an+1\right)}^2}+\dots +\frac{1}{{\left(an+n\right)}^2}\right)}\right)$
$\Rightarrow S=\underset{n\to \infty }{\lim }\frac{\sum _{r=1}^n{r}^{\frac{1}{3}}}{n^{\frac{7}{3}}\sum _{r=1}^n\frac{1}{{\left(na+r\right)}^2}}$
$S=\underset{n\to \infty }{\lim }\frac{\left(\sum _{r=1}^n{\left(\frac{r}{n}\right)}^{\frac{1}{3}}\frac{1}{n}\right)\times n^{\frac{1}{3}}}{n^{\frac{7}{3}}\sum _{r=1}^n\frac{1}{{\left(a+\frac{r}{n}\right)}^2}\frac{1}{n^2}}$
$S=\underset{n\to \infty }{\lim }\frac{\sum _{r=1}^n{\left(\frac{r}{n}\right)}^{\frac{1}{3}}.\frac{1}{n}}{\sum _{r=1}^n{\left(\frac{1}{a+\frac{r}{n}}\right)}^2\frac{1}{n}}$
$S=\frac{\underset{0}{\overset{1}{\int }}{x}^{\frac{1}{3}}dx}{\underset{0}{\overset{1}{\int }}\frac{1}{{\left(a+x\right)}^2}.dx}$
$S=\frac{{\left[\frac{x^{\frac{4}{3}}}{\frac{4}{3}}\right]}_0^1}{{\left[-\frac{1}{x+a}\right]}_0^1}$
$S=\frac{\frac{3}{4}}{\frac{1}{a}-\frac{1}{a+1}}=54$
$\frac{1}{a\left(a+1\right)}=\frac{1}{72}$
$a^2+a-72=0$
$a=8$or$a=-9$