无穷级数例子

$$计算\underset{x\to \infty }{\lim }(\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+...+\frac{1}{n+2n-1}+\frac{1}{n+2n})$$

解：

$$\begin{gathered} \underset{x\to \infty }{\lim }(\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+...+\frac{1}{n+2n-1}+\frac{1}{n+2n})= \\ \underset{x\to \infty }{\lim }\frac{1}{2n}(\frac{1}{\frac{1}{2}+\frac{1}{2n}}+\frac{1}{\frac{1}{2}+\frac{2}{2n}}+\frac{1}{\frac{1}{2}+\frac{3}{2n}}+...+\frac{1}{\frac{1}{2}+\frac{2n-1}{2n}}+\frac{1}{\frac{1}{2}+\frac{2n}{2n}})= \\ {\int }_o^1\frac{1}{\frac{1}{2}+x}dx=\ln (1/2+x){|}_0^1=\ln 3-\ln 2-\ln 1+\ln 2=\ln 3 \end{gathered}$$

以下做法错在哪里？

$$\begin{gathered} \underset{x\to \infty }{\lim }(\frac{1}{n+1}+\frac{1}{n+2}+\frac{1}{n+3}+...+\frac{1}{n+2n-1}+\frac{1}{n+2n})= \\ \underset{x\to \infty }{\lim }2\frac{1}{2n}(\frac{1}{1+\frac{1}{n}}+\frac{1}{1+\frac{2}{n}}+\frac{1}{1+\frac{3}{n}}+...+\frac{1}{1+\frac{2n-1}{n}}+\frac{1}{1+\frac{2n}{n}})= \\ 2{\int }_o^2\frac{1}{1+x}dx=2\ln (1+x){|}_0^2=2\ln 3 \end{gathered}$$