Example 4.28. Investigate the power series $\sum_{n=1}^{\infty} \frac{n^{2} x^{n}}{n!}$.
Solution. Here $a_{n}=\frac{n^{2}}{n!}$. By the D'Alembert's formula (Theorem 4.23):

$$
R=\lim _{n \rightarrow \infty} \frac{n^{2}}{n!}: \frac{(n+1)^{2}}{(n+1)!}=\lim _{n \rightarrow \infty} \frac{n^{2}(n+1)}{(n+1)^{2}}=+\infty
$$

Therefore, this series converges for all values of $x$. The convergence will be absolute and uniform in any interval $[-A, A], A>0$.