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.
Nothing. $S$ is not absolutely convergent, and that's it.
For one thing, for a sequence which is convergent but not absolutely convergent, the limit is dependent on the order of the terms. For an absolutely convergent series, you can reorder terms freely, extract different subseries and sum them separately, and so on. In other words, absolute convergence allows us to do many of the manipulations that we might want to do without having to be too careful about the infinity of the series.