Example 4.27. Investigate the power series $\sum_{n=1}^{\infty} \frac{(x+2)^{n}}{n^{n}}$.
Solution. We apply the Cauchy formula (Theorem 4.23) to determine the radius of convergence $R$:

$$
R=\frac{1}{\varlimsup_{n \rightarrow \infty} \sqrt[n]{\frac{1}{n^{n}}}}=\frac{1}{\varlimsup_{n \rightarrow \infty} \frac{1}{n}}=+\infty
$$

This result means that the power series converges for all values of $x$.