TS EAMCET · Maths · Definite Integration
\(\lim _{n \rightarrow \infty}\left[\frac{n}{n^2+1}+\frac{n}{2^2+n^2}+\ldots+\frac{1}{2 n}\right]=\)
- A \(\frac{\pi}{4}\)
- B \(\log 2\)
- C \(0\)
- D \(1\)
Answer & Solution
Correct Answer
(A) \(\frac{\pi}{4}\)
Step-by-step Solution
Detailed explanation
Let \(S_n=\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\ldots+\frac{1}{2 n}\) \(=\sum_{r=1}^n \frac{n}{n^2+r^2}=\sum_{r=1}^n \frac{1}{n\left(1+\frac{r^2}{n^2}\right)}\) Hence,…
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