WBJEE · Maths · Limits
\(\lim _{n \rightarrow \infty} \frac{\sqrt{n}}{\sqrt{\left(n^{3}\right)}}+\frac{\sqrt{n}}{\sqrt{(n+4)^{3}}}+\frac{\sqrt{n}}{\sqrt{(n+8)^{3}}}+\cdots \cdots \cdots+\frac{\sqrt{n}}{\sqrt{[n+4(n-1)]^{3}}}\) is
- 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 & Solution
Correct Answer
(A) \(\frac{5-\sqrt{5}}{10}\)
Step-by-step Solution
Detailed explanation
\(\lim _{n \rightarrow \infty} \sum_{r=0}^{n-1} \frac{\sqrt{n}}{\sqrt{(n+4 r)^{3}}}\) \(=\sum_{r=0}^{n-1} \frac{1}{n}\left(\frac{n \sqrt{n}}{\sqrt{(n+4 r)^{3}}}\right)\) \(=\sum_{r=0}^{n-1} \frac{1}{n}\left(\frac{1}{\left(1+\frac{4 r}{n}\right)^{3 / 2}}\right)\)…
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