Absolute convergence of sum and convergence to $0$ of product imply that at least one term in the product is $0$

Question

I'm trying to understand infinite products, and one of the sources I'm reading through says (without proof) that if $\sum |a_n|$ converges and $\prod (1+a_n)=0$ then $a_k=-1$ for some $k$. Why is this true?

Answer 1

$$\sum_{n=0}^{\inf}|a_{n}|$$converges means $$a_{n}$$goes to zero so if $$\prod(1+a_{n})$$ goes to zero then $$ln(\prod(1+a_{n}))=\sum ln(1+a_{n})$$ does not converge but since $$a_{n}$$ goes to zero$$ln(1+a_{n})$$ goes to zero but that would imply that the sum $$\sum ln(1+a_{n})$$ converges but we just know it can't and the only way this is possible is if one of the $$a_{n}$$ is -1 since $$ln(1-1)=ln(0)=\inf$$