What you are asking is exactly how to prove a particular case of the Riemann rearrengement theorem, that is if a series is convergent but the series of its abolute values diverge then, for each there exists a permutation such that .
In this especific case, , you have that both and are divergent (in the general case you would have to take the positive and the negative terms). Now, if is positive choose such that . Now define for . Again since is divergent to you can take such that . Take for . Now you use
are divergent to "move after A and before A" recursively. The key to show that the proces leads is that goes to 0, then passing to before A to after A or the converse will be "with a such small step as desired" for a sufficient number of repetitions.
This is just a sketch of Riemann adapted to your case, the idea is simple, but writting with precision is a little bit harder.