First, can an alternating series diverge? Or does it always converge since you're adding then subtracting? If it must converge, then some of the following may be moot.
Next, to review, if S is alternating, then we can examine |S|. We know that S < |S| since S is alternating and subtracts alternate terms. So, if |S| converges, we know that S must also converge, since S < |S|.
Case 1: S = converges........ |S| = converges..... Ergo, S is absolutely convergent.
Case 2: S = converges..... |S| = diverges...... Ergo, S is conditionally convergent.
But, what about this?
Case 3: S = unknown |S| = diverges...... What can we conclude about S ?
Also, if you know S is convergent, why go further with |S| in the first place? what is the point of examining |S| and adding the label S = "absolutely convergent"? It seems the entire point of absolute convergence, is to make a conclusion on the original S, if that is unknown.