Let $G$ be open in $\mathbb{C}$ and $f_n:G\rightarrow \mathbb{C}$ be a sequence of nonconstant holomorphic functions such that $\sum_{n=0}^\infty |f_n|$ converges uniformly on every compact subset of $G$.
Then, is $\prod_{n=0}^\infty (1+f_n)$ nonconstant?
I think the answer is yes, but I don't know how to prove this. How do I show this? Thank you in advance:)
Take nonconstant $f_n$ such that for for each $k$, $(1+f_{2k-1})(1+f_{2k}) = 1$. You should be able to pick $f_n$ small enough such that $\sum |f_n|$ converges uniformly (on compact sets) (multiply by $2^{-k}$ for example). The even partial products are all constantly one, so the limit is constant, one. I guess you could argue that I'm cheating a bit and if I group the even and odd guys I'm making a product of constant guys, but you could generalize the construction.