KCET · Maths · Limits
\(\lim _{n \rightarrow \infty}\left(\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\frac{n}{n^2+3^2}+\ldots+\frac{1}{5 n}\right)=\)
- A \(\pi / 4\)
- B \(\tan ^{-1} 3\)
- C \(\tan ^{-1} 2\)
- D \(\pi / 2\)
Answer & Solution
Correct Answer
(C) \(\tan ^{-1} 2\)
Step-by-step Solution
Detailed explanation
\(\lim _{n \rightarrow \infty}\left(\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\frac{n}{n^2+3^2}+\ldots+\frac{1}{5 n}\right)\)
\(=\lim _{n \rightarrow \infty}\left(\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\frac{n}{n^2+3^2}+\ldots+\frac{n}{n^2+(2 n)^2}\right)\)
\(=\lim _{n \rightarrow \infty} \sum_{r=1}^{2 n} \frac{n}{n^2+r^2}\)
\(=\lim _{n \rightarrow \infty} \sum_{r=1}^{2 n} \frac{1}{n}\left(\frac{1}{1+\left(\frac{r}{n}\right)^2}\right)\)
\(=\int_0^2 \frac{d x}{1+x^2}=\tan ^{-1}(2)\)
\(=\lim _{n \rightarrow \infty}\left(\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\frac{n}{n^2+3^2}+\ldots+\frac{n}{n^2+(2 n)^2}\right)\)
\(=\lim _{n \rightarrow \infty} \sum_{r=1}^{2 n} \frac{n}{n^2+r^2}\)
\(=\lim _{n \rightarrow \infty} \sum_{r=1}^{2 n} \frac{1}{n}\left(\frac{1}{1+\left(\frac{r}{n}\right)^2}\right)\)
\(=\int_0^2 \frac{d x}{1+x^2}=\tan ^{-1}(2)\)
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