Find $z_n$ divergent, so that $|z_n|$ converges?

Question

Do you know a divergent series or sequence $(z_n)_{n\in\mathbb{N}}$ ($z_n\in\mathbb{C}$), which absolut value $(|z_n|)_{n\in\mathbb{N}}$ converges?

I was not able to find one... only in the other direction (convergent becomes divergent), e.g. $\sum_n(-1)\frac{1}{n}$.

Answer 1

If you actually meant sequence, try $\;z_n:=(-1)^n\;{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$ ....

Answer 2

We know that absolute convergence of a series (in Banach space) implies convergence. Hence you cannot find a series which is absolutely convergent and not convergent.