I saw some place where someone wrote $\frac{1}{1-f(x)} = \sum_{n=0}^{\infty} f(x)^n, \forall x \in \mathbb{C}, $ where $f$ is some function s.t. $\vert f(x)\vert < 1, \forall x \in \mathbb{C}$.
Then he proved that the RHS $\sum_{n=0}^{\infty} f(x)^n$ absolutely converges $, \forall x \in \mathbb{C}$, in order that the RHS is well-defined and finite. I was wondering why we have to have the absolute convergence? Isn't that $\vert f(x)\vert < 1, \forall x \in \mathbb{C}$ can guarantee $\frac{1}{1-f(x)} = \sum_{n=0}^{\infty} f(x)^n, \forall x \in \mathbb{C}$?
I admit I am not familiar with complex analysis, and I am not sure if there is also a similar situation in Real analysis? I would appreciate if someone could point out what kinds of materials (such as Wikipedia articles or other internet links) will help me in this regard, besides explanation. Thanks for clarification!