# 单调有界定理与两个重要极限（二）

$\lim_{n\to\infty}(1+\frac{1}{n})^{n}=e$

\begin{align*}x_n =\left(1+\frac{1}{n}\right)^n &= 1+n\cdot \frac{1}{n}+\frac{n(n-1)}{1\cdot 2} \left(\frac{1}{n}\right)^2+\cdots+\frac{n!}{n!} \left(\frac{1}{n}\right)^{n} \\ &=1+1+\frac{1}{2!}\cdot\frac{n(n-1)}{n^2}+\cdots+\frac{1}{n!}\cdot\frac{n!}{n^n}\\ &=1+1+\frac{1}{2!}(1-\frac{1}{n})+\cdots+\frac{1}{n!}(1-\frac{1}{n})(1-\frac{2}{n})\cdots(1-\frac{n-1}{n})\end{align*}

\begin{align*}x_{n+1} =\left(1+\frac{1}{n+1}\right)^{n+1} &= 1+(n+1)\cdot \frac{1}{n+1}+\frac{(n+1)n}{1\cdot 2} \left(\frac{1}{n+1}\right)^2+\cdots+\frac{(n+1)!}{(n+1)!} \left(\frac{1}{n+1}\right)^{n+1} \\ &=1+1+\frac{1}{2!}\cdot\frac{n(n+1)}{(n+1)^2}+\cdots+\frac{1}{(n+1)!}\cdot\frac{(n+1)!}{(n+1)^{n+1}}\\ &=1+1+\frac{1}{2!}(1-\frac{1}{n+1})+\cdots+\\ &\quad +\frac{1}{(n+1)!}(1-\frac{1}{n+1})(1-\frac{2}{n+1})\cdots +(1-\frac{n}{n+1})\end{align*}

\begin{align*}x_n = &1+1+\frac{1}{2!}(1-\frac{1}{n})+\cdots+\frac{1}{n!}(1-\frac{1}{n})(1-\frac{2}{n})\cdots(1-\frac{n-1}{n})\\ &\le 1+1+\frac{1}{2!}+\frac{1}{3!}\cdots+\frac{1}{n!}\\ &\le 1+1+\frac{1}{2}+\frac{1}{2^2}\cdots+\frac{1}{2^{n-1}} \\ &=1+\frac{1-\frac{1}{2^n}}{1-\frac{1}{2}}\le 3\end{align*}

$\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n=e$