证明,对任意的正整数都有
$$\frac{1}{n+1}<\ln(1+\frac{1}{n})<\frac{1}{n}$$

证明:
$$\ln(1+\frac{1}{n})=\ln(\frac{n+1}{n})=\ln(n+1)-\ln(n)$$
根据拉格朗日中值定理,有

$$\ln(n+1)-\ln(n)=(n+1-n)*\frac{1}{\xi}=\frac{1}{\xi} ~~~ (n<\xi<{n+1})$$

$$\therefore \frac{1}{n+1}<\ln(1+\frac{1}{n})<\frac{1}{n}$$