Question

$\underset{n\to \infty }{\lim }\frac{\sqrt{n}}{\sqrt{\left(n^3\right)}}+\frac{\sqrt{n}}{\sqrt{(n+4{)}^3}}+\frac{\sqrt{n}}{\sqrt{(n+8{)}^3}}+\dots ..+\frac{\sqrt{n}}{\sqrt{[n+4(n-1){]}^3}}$

• 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)

$\underset{n\to \infty }{\lim }\sum _{r=0}^{n-1}\frac{\sqrt{n}}{\sqrt{(n+4r{)}^3}}$
$=\sum _{r=0}^{n-1}\frac{1}{n}\left(\frac{n\sqrt{n}}{\sqrt{(n+4r{)}^3}}\right)$
$=\sum _{r=0}^{n-1}\frac{1}{n}\left(\frac{1}{{\left(1+\frac{4r}{n}\right)}^{3/2}}\right)$
$=\underset{0}{\overset{1}{\int }}\frac{dx}{(1+4x{)}^{3/2}}$
$=\frac{1}{4}\underset{1}{\overset{5}{\int }}\frac{dz}{z^{3/2}}={\left(\frac{1}{4}\left(\frac{-2}{\sqrt{z}}\right)\right)}_1^5$
$=\frac{5-\sqrt{5}}{10}$