Example 4.14. Investigate the series $\sum_{n=1}^{\infty}(-1)^{n} \frac{n!}{n}$.
Solution. According to Theorem 4.9

$$
\lim _{n \rightarrow \infty} \sqrt[n]{\frac{n!}{n^{n}}}=\lim _{n \rightarrow \infty} \frac{\sqrt[n]{n!}}{n}=\lim _{n \rightarrow \infty} \frac{(2 \pi n)^{1 / 2 n} \frac{n}{e} e^{\frac{\Theta}{12 n}}}{n}=\frac{1}{e}<1
$$

Therefore, the series converges absolutely.