\(\displaystyle \lim_{n\to \infty}\left[\frac{n}{1^2+n^2}+\frac{n}{2^2+n^2}+\ldots+\frac{n}{n^2+n^2}\right]\)

答案是π/4不知道該用什麼方法切入
麻煩大家提供一些想法～

\(\displaystyle \lim_{n\to\infty}\sum_{k=1}^n\frac{n}{k^2+n^2}\)

\(\displaystyle =\lim_{n\to\infty}\sum_{k=1}^n\frac{1}{\left(\frac{k}{n}\right)^2+1}\frac{1}{n}\)

\(\displaystyle =\int_0^1\frac{1}{x^2+1}dx\)

（令 \(x=\tan\theta\Rightarrow dx=\sec^2\theta d\theta\)）

\(\displaystyle =\int_0^{\frac{\pi}{4}}\frac{1}{\tan^2\theta+1}\cdot\sec^2\theta d\theta\)

\(\displaystyle =\int_0^{\frac{\pi}{4}}1 d\theta\)

\(\displaystyle =\frac{\pi}{4}\)

謝謝～