Convergence of $\prod_{n=1}^\infty(1+a_n)$ The question is motivated by the following exercise in complex analysis:

Let $\{a_n\}\subset{\Bbb C}$ such that $a_n\neq-1$ for all $n$. Show that if $\sum_{n=1}^\infty |a_n|^2$ converges, then the product $\prod_{n=1}^\infty(1+a_n)$ converges to a non-zero limit if and only if $\sum_{n=1}^\infty a_n$ converges.

One can get a proof by using $|a_n|^2$ to bound $|\log(1+a_n)-a_n|$. 
Here is my question: is the converse of this statement also true?

If "the product $\prod_{n=1}^\infty(1+a_n)$ converges to a non-zero limit if and only if $\sum_{n=1}^\infty a_n$ converges", then $\sum_{n=1}^\infty |a_n|^2$ converges.


The convergence of $\Pi_{n=0}^\infty(1+a_n)$ is equivalent to that of $\sum_{n=0}^\infty\ln(1+a_n)$. Note 
$$ \ln(1+x)=x-\frac{1}{2}x^2+O(|x|^3). $$
So
$$\sum_{n=0}^\infty\ln(1+a_n)=\sum_{n=0}^\infty (a_n-\frac{1}{2}a_n^2+O(|a_n|^3))=\sum_{n=0}^\infty a_n-\frac{1}{2}\sum_{n=0}^\infty a_n^2+\sum_{n=0}^\infty O(|a_n|^3).$$
Note the convergence of $\sum_{n=0}^\infty|a_n|^2$ implies the convergence of $\sum_{n=0}^\infty a_n^2$ and $\sum_{n=0}^\infty O(|a_n|^3)$. Thus the product $\Pi_{n=0}^\infty(1+a_n)$ converges to a non-zero limit if and only if $\sum_{n=0}^\infty a_n$ converges.
