That's the idea indeed. Maybe you could take an explicit example .
The question is if the absolute value of a sequence converges then does the sequence itself converge. My gut reaction to this is no because you could have something with opposite signs (+ - + -) that would converge in absolute value but diverge otherwise, or even converge to different values. I know that lim n-> infinity |n/-n|=1 and n/-n =-1 but they both converges. No luck on the convergent, divergent scenario. (Keep in mind this is just some random idea about this problem from my head). What do you think?
A perfect example is the alternating harmonic series. It is convergent and has a value of , but it is not absolutely convergent, because the series of absolute values is the harmonic series, which is divergent.