\[{\text{Find }}\mathop {\lim }\limits_{n \to + \infty } \frac{n}{{\sqrt[n]{{n!}}}}.\]
\[\begin{array}{l}
\ln \frac{n}{{\sqrt[n]{{n!}}}} = \frac{{n\ln n - \ln n!}}{n} \hfill \\
{\text{Then apply Stolz theorem}}{\text{, }}{x_n} = n\ln n - \ln n!,{\text{ }}{y_n} = n. \hfill \\
\end{array} \]
\[e\]
Теорема
§
$$\eqalign{
{\text{Если}} \hfill \\
{\text{а) }}{y_{n + 1}} > {y_n}, \hfill \\
{\text{б) }}\mathop {\lim }\limits_{n \to \infty } {y_n} = + \infty , \hfill \\
{\text{в) существует }}\mathop {\lim }\limits_{n \to \infty } \frac{{{x_{n + 1}} - {x_n}}}{{{y_{n + 1}} - {y_n}}}, \hfill \\
{\text{то}} \hfill \\
\mathop {\lim }\limits_{n \to \infty } \frac{{{x_n}}}{{{y_n}}} = \mathop {\lim }\limits_{n \to \infty } \frac{{{x_{n + 1}} - {x_n}}}{{{y_{n + 1}} - {y_n}}}. \hfill \\
} $$