Consider an infinite product $$p=\prod_{n=1}^\infty a_n,$$ with $a_n\in\mathbb{R}$ (or $\mathbb{C}$ if possible). Is there an if and only if type theorem for when $p=1$, or is anything known about the nature of the $a_n$ when $p=1$ ?
Clearly one case springs to mind: $a_n=1$ for all $n\geq 1$. Is this the only case or are there others?
Additionally: What if the $a_n$ were monotonically increasing or decreasing?
I am writing my comment here since I have not enough reputation to leave a comment. I am not sure on my response. By taking logarithm from both sides we have this:
$\sum_{i=1}^{\infty}b_i=0$ in which $b_i=\log{a_i}$
Now we have an infinite summation which is equal to zero and it could have any arrangement. I think we can not say any thing about this since we can have several arrangement that yields the situation. If we consider the situation for finite elements there are several occasions in which the summation is zero and there is not any further relation on elements.