I have been working on this problem for a while and cannot seem to make any progress without coming up with something wrong or hitting a dead end.
Here is what I have so far:
$ \prod (1+a_n) < \infty \implies \sum a_n < \infty $: Similarly we ignore finitely many terms until $|a_n| \leq 1/2$ and we use the taylor series for the product. We have that $\prod 1+ a_n$ converging im plying that $\sum \log (1+a_n)$ converges to a nonzero limit since none of the factors are 0 as $a_n \neq -1$. We have that \begin{eqnarray} |\sum \log(1+a_n) | =| \sum (a_n-a_n^2/2 +\ldots) | \\ \geq | \sum (a_n-|a_n^2/2 +\ldots|) | \geq \left| \sum a_n-|a_n|^2-|a_n|^3-\ldots \right| \\ = \left|\sum a_n-|a_n|^2(1+|a_n|+|a_n|^2+|a_n|^3+\ldots \right| \end{eqnarray} by the triangle inequality. Thus $\infty > |\sum \log(1+a_n) | \geq \left|\sum a_n-2|a_n|^2 \right|$ from the previous part. Thus $\left|\sum a_n-2|a_n|^2 \right|$ is convergent, and since $\sum|a_n|^2$ is absolutely convergent we can split the series (I don't really know if this is even true) and we have that partial sums $|\sum a_n|$ is bounded.
Any help would be appreciated! Also any good references for getting better at this kind of stuff would be great!!