WBJEE · Maths · Definite Integration
The value of \(\lim _{n \rightarrow \infty}\left\{\frac{\sqrt{n+1}+\sqrt{n+2}+\ldots+\sqrt{2 n-1}}{n^{3 / 2}}\right\}\) is
- A \(\frac{2}{3}(2 \sqrt{2}-1)\)
- B \(\frac{2}{3}(\sqrt{2}-1)\)
- C \(\frac{2}{3}(\sqrt{2}+1)\)
- D \(\frac{2}{3}(2 \sqrt{2}+1)\)
Answer & Solution
Correct Answer
(A) \(\frac{2}{3}(2 \sqrt{2}-1)\)
Step-by-step Solution
Detailed explanation
\(\because \lim _{n \rightarrow \infty \left[\frac{\sqrt{n+1}+\sqrt{n+2}+\ldots+\sqrt{2 n-1}}{n^{\frac{3}{2}}}\right]}\) \(=\lim _{n \rightarrow \infty 1}\left[\sqrt{1+\frac{1}{n}}+\sqrt{1+\frac{2}{n}}+\ldots+\sqrt{1+\frac{n-1}{n}}\right] \frac{1}{n}\)…
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