The series $\sum_{n=1}^\infty(z^{2^n}-z^{-2^{n}})^{-1}$ converges compactly in $\mathbb{C} \setminus(\{0\} \cup \mathbb{D} \})$

Question

I need to show that the series $\sum_{n=1}^\infty(z^{2^n}-z^{-2^{n}})^{-1}$ converges uniformly on compact subsets of $\mathbb{C} \setminus(\{0\} \cup \mathbb{D} \}$, where $\mathbb{D} = \{ z \in \mathbb{C}: |z|=1\}$.Can you help me?

Answer 1

For $1+\epsilon \leq |z| \leq M$ use $|z^{2^{n}}-z^{-2^{n}}| \geq (1+\epsilon)^{2{n}}-M^{-2^{n}}$ and use M-test. For $\epsilon \leq |z| \leq 1-\epsilon$ just replace $z$ by $\frac 1 z$ in the first case. Any compact set in the region is contained in one of these closed annulii for suitable $\epsilon$ and $M$..