Q108 / 155 · WBJEE 2021 · Mathematics

$\lim _{n \rightarrow \infty} \frac{\sqrt{n}}{\sqrt{\left(n^{3}\right)}}+\frac{\sqrt{n}}{\sqrt{(n+4)^{3}}}+\frac{\sqrt{n}}{\sqrt{(n+8)^{3}}}+\cdots \cdots \cdots+\frac{\sqrt{n}}{\sqrt{[n+4(n-1)]^{3}}}$ is

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

Solution

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

Hence, the correct answer is option: 'A'.

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