Example 8. Find the region of convergence of the series

$$
\sum_{n=1}^{\infty} \frac{(1+i)^{n+1}}{z^{n}}
$$
Solution. Here $c_{-n}=(1+i)^{n+1}, c_{-n-1}=(1+i)^{n+2}, z_{0}=0$. Therefore,

$$
r=\lim _{n \rightarrow \infty} \frac{\left|(1+i)^{n+2}\right|}{\left|(1+i)^{n+1}\right|}=\lim _{n \rightarrow \infty}|1+i|=\sqrt{2}
$$

The series converges in the region $|z|>\sqrt{2}$.