If and are convergent then (no absolute convergence is used here).
If or are divergent then it does not even make sense to write . Because what is that even supposed to mean?
There is no rearrangment going on over here (in what Moo did). Just term by term addition.For you to be able to manipulate a series in the manner you are referring to it must be Cesaro Summable, and one of those conditions is that the series is convergent.
An example where rearanging is not permissable is the following
Let and be convergent series. Then and are convergent sequences. Now if and are convergent it means is a convergent. Furthermore, . By definition, and . And . Thus, .
That is easy. If you combine something that is finite and infinite you get something that is infinite. So what is the problem?
Just joking! That is how they "prove" in physics courses. Say is convergent and is divergent. And assume that is convergent. Then it would mean is convergent. A contradiction!