Analyticity at infinity: Laurent series

Question

Suppose that the series $$A_0 + A_1(z-z_0)^{-1} + \cdots + A_n (z-z_0)^{-n}+ \cdots$$ converges to $f(z)$ for all $z$ such that $r<|z-z_0|<\infty$ and let $f^*(\xi)=f(1/\xi), f^*(0)=A_0$. Show that $f^*$ is analytic at $\xi=0$. The hint I have is that if $z_0 \neq0$ consider the series $$f^*(\xi)= \sum_{n=0}^{\infty}A_n \left( \frac{\xi}{1- \xi z_0} \right)^n$$ where $|\xi|<min \lbrace r,|z_0|^{-1} \rbrace$.
I don't know how to use the above to prove that $f$ is analytic at $\infty$. Any hint can be useful.
Thanks.