lim[(n!)^(1/n)]/n的極限

求如下極限 $\lim\limits _{n \rightarrow +\infty} \frac{\sqrt[n]{n !}}{n} \text { 式 }(1)$
對式①取對數得
$\begin{aligned} &\lim _{n \rightarrow +\infty} \ln \sqrt[n]{\frac{n !}{n^{n}}}\\ &=\frac{1}{n} \lim _{n \rightarrow +\infty} \ln \frac{n !}{n^{n}}\\ &=\frac{1}{n} \lim _{n \rightarrow +\infty}\left(\ln \frac{n}{n}+\ln \frac{n-1}{n}+\cdots+\ln \frac{1}{n}\right)\\ &=\frac{1}{n} \lim _{n \rightarrow +\infty} \sum_{i=1}^{n} \frac{i}{n} \text { 式(2)} \end{aligned}$
由定積分定義得式②
$\begin{aligned} &=\int_{0}^{1} \ln x d x\\ &=\left.x \ln x\right|_{0} ^{1}-\int_{0}^{1} x d \ln x \\ &=[x \ln x-x]_{0}^{1}=-1 \end{aligned}$
即 $\lim\limits _{n \rightarrow +\infty} \ln \sqrt[n]{\frac{n !}{n^{n}}}=-1$
$\Rightarrow \lim\limits _{n \rightarrow +\infty} \frac{\sqrt[n]{n !}}{n}=\frac{1}{e}$