Example 5.18. Find the region of convergence of the power series

$$
x+2!x^{2}+3!x^{3}+\ldots+n!x^{n}+\ldots
$$
Solution. Here $u_{n}=n!|x|^{n}, u_{n+1}=(n+1)!|x|^{n+1}$.

From this,

$$
l=\lim _{n \rightarrow \infty} \frac{u_{n+1}}{u_{n}}=\lim _{n \rightarrow \infty} \frac{(n+1)!|x|^{n+1}}{n!|x|^{n}}=|x| \lim _{n \rightarrow \infty}(n+1)
$$

Thus,

$$
l=\begin{array}{ccc}
\infty & \text { when } & x \neq 0, \\
0 & \text { when } & x=0 .
\end{array}
$$

Therefore, according to the D'Alembert's ratio test, the series converges only at the point $x=0$.