📝 题目
例 10 求 $\displaystyle{\mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{{2}^{n}n!}{{n}^{n}}}$ .
💡 答案与解析
解 令 ${x}_{n} = \frac{{2}^{n}n!}{{n}^{n}}$ ,则有 $\frac{{x}_{n + 1}}{{x}_{n}} = \frac{2}{{\left( 1 + \frac{1}{n}\right) }^{n}} \leq 1 \Rightarrow {x}_{n} \downarrow$ . 又
$$ {x}_{n} > 0 \Rightarrow \exists \mathop{\lim }\limits_{{n \rightarrow \infty }}{x}_{n} $$
设 $\displaystyle{\mathop{\lim }\limits_{{n \rightarrow \infty }}{x}_{n} = a}$ ,再注意到 ${x}_{n + 1} = \frac{2{x}_{n}}{{\left( 1 + \frac{1}{n}\right) }^{n}}$ ,两端取极限得到
$$ a = \frac{2}{\mathrm{e}}a \Rightarrow a = 0,\;\text{ 即 }\;\mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{{2}^{n}n!}{{n}^{n}} = 0. $$