JEE Mains · Maths · STD 12 - 7.2 definite integral
The value of \(\lim _{n \rightarrow \infty} \sum_{k=1}^n \frac{n^3}{\left(n^2+k^2\right)\left(n^2+3 k^2\right)}\) is :
- A \(\frac{(2 \sqrt{3}+3) \pi}{24}\)
- B \(\frac{13 \pi}{8(4 \sqrt{3}+3)}\)
- C \(\frac{13(2 \sqrt{3}-3) \pi}{8}\)
- D \(\frac{\pi}{8(2 \sqrt{3}+3)}\)
Answer & Solution
Correct Answer
(B) \(\frac{13 \pi}{8(4 \sqrt{3}+3)}\)
Step-by-step Solution
Detailed explanation
\( \lim _{n \rightarrow \infty} \sum_{k=1}^n \frac{n^3}{n^4\left(1+\frac{k^2}{n^2}\right)\left(1+\frac{3 k^2}{n^2}\right)} \) \( =\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^n \frac{n^3}{\left(1+\frac{k^2}{n^2}\right)\left(1+\frac{3 k^2}{n^2}\right)}\)…
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