Uniform convergence of sum from entire infinite product

Question

For $\{f_n\}$ being holomorphic functions, we know that $\sum_{n=1}^\infty |f_n(z)|$ converges uniformly implies that the infinite product $\prod_{n=1}^\infty (1+f_n(z))$ converges uniformly as well, and so the product is holomorphic on $\mathbb{C}$. Does $\prod_{n=1}^\infty (1+f_n(z))$ being entire also imply that $\sum_{n=1}^\infty |f_n(z)|$ converges uniformly? I do not think my proof for the first statement works for this one, so I'm a bit curious as to whether or not this holds.

Answer 1

No. For a somewhat contrived example, take $f_n(z)$ to be the constant function $a_n$ where $a_n = (-1)^n/n$ or some other slowly decreasing alternating sequence. Then $\prod_{n=1}^\infty (1+f_n(z))$ is convergent (hence entire) but $\sum_{n=1}^\infty |f_n(z)|$ does not converge anywhere.