Convergence of an infinite product to an element of $\mathbb{C} \setminus \mathbb{R}_{-}$

Question

Let $(a_n)_{n\geq 1} \subset \mathbb{C}$ a sequence such that $1 + a_n \in \mathbb{C}\setminus\mathbb{R}_{-}$. Suppose that the infinite product \begin{align} \prod_{n=1}^{\infty} (1 + a_n) \end{align} converges, i.e. \begin{align} p_N : = \prod_{n=1}^{N} (1+ a_n) \xrightarrow[N\to\infty]{} \, \prod_{n=1}^{\infty} (1+a_n) =: p \neq 0 \end{align}

I have to show that $p\in \mathbb{C}\setminus\mathbb{R}_{-}$ and $p_N \in \mathbb{C}\setminus\mathbb{R}_{-}$.

I know that the sequence $(a_n)$ converges to 0, but I don't see how to use this fact to prove what required. I tried to show that the series of the arguments is not equal to $\pi$ or $-\pi$, but without success.

Any suggestions? Thanks in advance!

Answer 1

Consider the sequence: $a_1=i-1, a_2=i-1$ and $a_n=0$ for $n>2$. Then $1+a_n \in \mathbb{C} \setminus \mathbb{R}_{-}$, but: $\prod_{n=1}^\infty 1+a_n=i^2=-1$.

Answer 2

Let $a_{2n-1}=i/n$ and $a_{2n}=-1+(1+a_{2n-1})^{-1}.$ Then no $a_n$ is real but the infinite product is $1.$