Example 3. Investigate the convergence of the power series $\sum_{n=0}^{\infty} \frac{x^{n}}{n!}$.
Solution. Let's determine the radius of convergence of the power series using D'Alembert's formula. We have (it is customary to consider $0!=1$):

$$
R=\lim _{n \rightarrow \infty}\left|\frac{a_{n}}{a_{n+1}}\right|=\lim _{n \rightarrow \infty} \frac{(n+1)!}{n!}=\lim _{n \rightarrow \infty}(n+1)=+\infty
$$

Answer. The given series converges absolutely for all $x \in \mathbb{R}$.