Example 5.8. Using the D'Alembert's criterion, investigate the convergence of the series

$$
\frac{1}{10}+\frac{1 \cdot 2}{10^{2}}+\frac{1 \cdot 2 \cdot 3}{10^{3}}+\ldots+\frac{n!}{10^{n}}+\ldots
$$
Solution. Knowing $a_{n}=\frac{n!}{10^{n}}$, let's find the $(n+1)$-th term of the series: $a_{n+1}=\frac{(n+1)!}{10^{n+1}}$. We will compute

$$
\lim _{n \rightarrow \infty} \frac{a_{n+1}}{a_{n}}=\lim _{n \rightarrow \infty} \frac{(n+1)!}{10^{n+1}}: \frac{n!}{10^{n}}=\lim _{n \rightarrow \infty} \frac{(n+1)!10^{n}}{n!10^{n+1}}=\lim _{n \rightarrow \infty} \frac{n+1}{10}=\infty .
$$

Since $l=\infty>1$, the series diverges.