\[\begin{align} &\lim_{n\rightarrow \infty}\frac{1}{n}(\sqrt{1+\cos{\frac{\pi}{n}}}+\sqrt{1+\cos{\frac{2\pi}{n}}}+...+\sqrt{1+\cos{\frac{n\pi}{n}}})\\ &解答：I_n=\frac{1}{n}\sum_{k=1}^{n}\sqrt{1+\cos\frac{k\pi}{n}}\\ &\lim_{n\rightarrow \infty}a_n=\lim_{n\rightarrow \infty}\frac{1}{n}\sum_{k=1}^{n}{\sqrt{1+\cos{\frac{k\pi}{n}}}}\leq I_n \leq \frac{1}{\pi}\lim_{n\rightarrow \infty}\frac{1}{n}\sum_{k=1}^{n}{\sqrt{1+\cos\frac{k\pi}{n}}}=\lim_{n\rightarrow \infty}b_n\\ &\lim_{n\rightarrow \infty}a_n=\lim_{n\rightarrow \infty}\frac{1}{n}\sum_{k=1}^{n}{\sqrt{1+\cos{\frac{k\pi}{n}}}}=\int_0^1{\sqrt{1+\cos x}dx=\frac{2\sqrt2}{\pi}}\\ &\lim_{n\rightarrow \infty}b_n=\frac{1}{\pi}\lim_{n\rightarrow \infty}\frac{1}{n}\sum_{k=1}^{n}{\sqrt{1+\cos{\frac{k\pi}{n}}}}=\frac{1}{\pi}\int_0^\pi\sqrt{1+\cos x}dx=\frac{2\sqrt2}{\pi}\\ &I_n=\frac{2\sqrt2}{\pi} \end{align} \]

\[\begin{align} &\lim_{n\rightarrow \infty}(\frac{\sin{\frac{\pi}{n}}}{n+1}+\frac{\sin{\frac{2\pi}{n}}}{n+2}+...+\frac{\sin{\frac{n\pi}{n}}}{n+\frac{1}{n}})\\ &解答：\int_0^1{f(x)}dx=\lim_{\lambda \rightarrow \infty }\sum_{i=1}^{n}{f(\xi_i)\Delta x_i}=\lim_{n\rightarrow \infty}\frac{1}{n}\sum_{i=1}^{n}f(\frac{i}{n})\\ &I_n=\sum_{k=1}^{n}\frac{1}{n}(\frac{\sin{\frac{k\pi}{n}}}{1+\frac{1}{kn}})\\ &a_n=\sum_{k=1}^{n}\frac{n}{n+1}\sin{\frac{k\pi}{n}}\frac{1}{n}\leq I_n \leq\sum_{k=1}^{n}\sin{\frac{k\pi}{n}}\frac{1}{n}=b_n\\ &\lim_{n\rightarrow \infty} a_n=\int_0^1{\sin(\pi x)}dx=\frac{2}{\pi}\\ &\lim_{n\rightarrow \infty} b_n=\int_0^1{\sin(\pi x)}dx=\frac{2}{\pi}\\ &I_n=\frac{2}{\pi}\\ \end{align} \]