Relation between the convergence of $\sum a_{n}$ and $\prod (1+a_{n})$

Question

What is the relation between the convergence of $\sum a_{n}$ and $\prod (1+a_{n})$ where $a_{n} \in \mathbb{C} \ \forall n$ ?

Where can I find some references about this topic ?

Answer 1

Let $a_n\neq 0$ for all $n\in \Bbb N$, then $$\prod_{n=1}^\infty a_n \text{ converges} \Leftrightarrow \sum_{n=1}^\infty \log{a_n} \text{ converges}$$ Moreover, for $a_n \neq -1$, we have an equivalence $$\sum_{n=1}^\infty \log(1+a_n) \text{ converges absolutely} \Leftrightarrow \sum_{n=1}^\infty a_n \text{ converges absolutely}.$$ Thus, if $\sum_{n=1}^\infty a_n$ converges absolutely, $\prod_{n=1}^\infty (1+a_n)$ converges unconditionally. Unfortunately, I have no reference for this in English language.

Answer 2

This is definitely true when $a_n$ are positive numbers as demonstrated here.

However, when $a_n \in \mathbb{C}$, then convergence of $\sum|a_n|$ is just a necessary condition for convergence of the infinite product.