问题及解答

求极限

\[\lim\limits_{n\rightarrow\infty}(\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\cdots+\frac{n}{n^2+(2n)^2})\]

\[
\begin{split}
\text{原式}&=\lim_{n\rightarrow\infty}\sum_{k=1}^{2n}\frac{n^2}{n^2+k^2}\cdot\frac{2}{2n}\\
&=\lim_{n\rightarrow\infty}\sum_{k=1}^{2n}\frac{1}{1+(\frac{k}{n})^2}\cdot\frac{1}{n}\\
&=\int_0^2\frac{1}{1+x^2}\mathrm{d}x\\
&=\arctan x\biggr|_0^2=\arctan 2.
\end{split}
\]