If we know the following convergence criteria for infinite products is correct
$$\boxed{\sum a_{n}\text{ converges }\Leftrightarrow\prod\left(1+a_{n}\right)\text{ converges, for real }a_{n}>0}$$
Then it immediately follows that
$$\boxed{\sum|a_{n}|\text{ converges }\Leftrightarrow\prod(1+|a_{n}|)\text{ converges}}$$
because $|a_n|>0$, even for complex $a_n$ as required by the first criteria.
Question: This this logic correct? I am asking because I have seen source which derive the first criterion (eg), and then derive the second in a complicated way. What am I missing?