I have below 2 very similar questions about divergence of series and sequences:
If $\sum a_n$ diverges, does $\sum |a_n|$ diverge?
If $\sum |a_n|$ diverges, does $\sum a_n$ diverge as well?
This would be sort of opposite to absolute convergence, and would be concerning "absolute divergence". For the first one, I think $\sum a_n$ will diverge, since I have not found a counterexample for it.
For the second one, let $a_n = \frac{(-1)^n}{n}$, then the absolute value of $\sum a_n$ diverges, while the actual sum $\sum a_n$ converges.
Best Answer
If a series converges absolutely, then it converges (you seem to be aware of that). That answers the first question (which is just the contrapositive).
As for the second, certain series converge conditionally, for example $$ 1-1/2+1/3-1/4+ etc. $$ which means precisely that they, although convergent, are not absolutely convergent. That answers your second question