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 ?

If I am not mistaken, I believe their convergence is equivalent, by taking the logarithm of the product. I'll see around for references. — Ian Mateus, Dec 30 '13 at 14:34

score 2 · Accepted Answer · answered Dec 30 '13 at 14:47

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.

score 1 · Answer 2 · answered Dec 30 '13 at 14:48

1

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.

answered Dec 30 '13 at 14:48

Sourav D

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

2 Answers2

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