Suppose $f(z)$ is analytic for $|z - z_0| > r$, then what is the series expansion of $f$? [closed]

Suppose $f(z)$ is analytic for $|z - z_0| > r$, then do we have that $f(1/z)$ is analytic for $|z - z_0| < r$ and the following series expansion? Why?

$$f(z) = \sum_{n=0}^{\infty} \frac{a_n}{(z - z_0)^n}, \quad |z - z_0| > r$$