I have a question about the proof of Lemma 11.22 in Iwaniec and Kowalski Analytic Number Theory.
The lemma states:
Let $w_1,\dots, w_r$ be complex numbers, $A$, $B$ positive real numbers and assume that $$ \Big|\sum_{j=1}^{r} w_j^n \Big| \le AB^n $$ holds for all integers $n$ large enough. Then $|w_j|\le B$ for all $j$.
The proof in the book is as follows. Consider the complex power series $$ f(z)=\sum_{n\ge 1}\Big(\sum_{j}w_j^n\Big)z^n=\sum_j\frac{1}{1-w_jz}. $$ The hypothesis implies that $f$ converges absolutely in the disc $|z|<1/B$, hence $f$ is analytic in this region. In particular, it has no poles there, which means that we must have $|w_j|^{-1}\ge B^{-1}$ for all $j$.
My question is: in the expression for $f$, why can we rearrange the sum as in the rightmost? (even when we restrict ourselves to the disc $|z|<1/B$.)