Question

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

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

Answer 2

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Answer 3

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