Question

$\lim \limits_{n \rightarrow \infty} \frac{\sqrt{n}}{\sqrt{\left(n^{3}\right)}}+\frac{\sqrt{n}}{\sqrt{(n+4)^{3}}}+\frac{\sqrt{n}}{\sqrt{(n+8)^{3}}}+\ldots . .+\frac{\sqrt{n}}{\sqrt{[n+4(n-1)]^{3}}}$

• A. $\frac{5-\sqrt{5}}{10}$
• B. $\frac{5+\sqrt{5}}{10}$
• C. $\frac{2+\sqrt{3}}{2}$
• D. $\frac{2-\sqrt{3}}{2}$

Answer 1

Answer: (A)

$\lim\limits _{n \rightarrow \infty} \displaystyle\sum_{r=0}^{n-1} \frac{\sqrt{n}}{\sqrt{(n+4 r)^{3}}}$
$=\displaystyle\sum_{r=0}^{n-1} \frac{1}{n}\left(\frac{n \sqrt{n}}{\sqrt{(n+4 r)^{3}}}\right)$
$=\displaystyle\sum_{r=0}^{n-1} \frac{1}{n}\left(\frac{1}{\left(1+\frac{4 r}{n}\right)^{3 / 2}}\right)$
$=\int\limits_{0}^{1} \frac{d x}{(1+4 x)^{3 / 2}}$
$=\frac{1}{4} \int\limits_{1}^{5} \frac{d z}{z^{3 / 2}}=\left(\frac{1}{4}\left(\frac{-2}{\sqrt{z}}\right)\right)_{1}^{5}$
$=\frac{5-\sqrt{5}}{10}$