absolute convergence of infinite product

Question

We know if $\sum_{n=1}^\infty|z_n|$ converges then $\sum_{n=1}^\infty z_n$ converges absolutely. (kind of trivial)

I wonder whether it holds for infinite products, that is, if $\prod_{n=1}^\infty|z_n|$ converges then $\prod_{n=1}^\infty z_n$ converges absolutely.

Is the statement above true or not?

Thanks a lot!

I know $\prod_{n=1}^\infty|z_n|$ converges doesn't imply $\prod_{n=1}^\infty z_n$ converges.

Answer 1

No. Take $z_n=-1$, so that $\prod_{n=1}^\infty |z_n|$ trivially converges to $1$. But $z_n=1+(-2)$, and the definition of $\prod_{n=1}^\infty (1+(-2))$ converging absolutely is that $\prod_{n=1}^\infty (1+|{-}2|) = \prod_{n=1}^\infty 3$ converges, which it does not.