JEE Mains · Maths · STD 12 - 7.2 definite integral
\(\lim _{n \rightarrow \infty}\left(\frac{n^{2}}{\left(n^{2}+1\right)(n+1)}+\frac{n^{2}}{\left(n^{2}+4\right)(n+2)}+\frac{n^{2}}{\left(n^{2}+9\right)(n+3)}+\ldots+\frac{n^{2}}{\left(n^{2}+n^{2}\right)(n+n)}\right)\) is equal to
- A \(\frac{\pi}{8}+\frac{1}{4} \log _{ e } 2\)
- B \(\frac{\pi}{4}+\frac{1}{8} \log _{ e } 2\)
- C \(\frac{\pi}{4}-\frac{1}{8} \log _{ e } 2\)
- D \(\frac{\pi}{8}+\log _{ e } \sqrt{2}\)
Answer & Solution
Correct Answer
(A) \(\frac{\pi}{8}+\frac{1}{4} \log _{ e } 2\)
Step-by-step Solution
Detailed explanation
\(\lim _{n \rightarrow \infty}\left(\sum \limits_{r=1}^{n} \frac{n^{2}}{\left(n^{2}+r^{2}\right)(n+r)}\right)\) \(=\lim _{n \rightarrow \infty}\left(\sum \limits_{r=1}^{n} \frac{1}{n\left(1+\left(\frac{r}{n}\right)^{2}\right)\left(1+\left(\frac{r}{n}\right)\right)}\right)\)…
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