KnifeMissile

Oh, whatever! You all seem to already know what a conditionally convergent series is, anyways. Yes, that was what I was trying to build up to. No physicist I have ever talked to has taken my word that such a thing exists which is why I thought you'd all find it fascinating. They all believe that addition is necessarily commutative. Oh well, I can still explain how such a thing exists. It won't be a formal proof but it won't need to be. As long as I can describe the idea in detail, you can all formlize it for yourselves (and you all sound like you can!).

So, if you are a conditionally convergent series, then the sum of your terms of like sign must diverge (in other words, the sum of all the negative terms must diverge negatively, while all positive terms diverge positively). Otherwise, you must be absolutely convergent. I mean, if only your negative terms diverge, how can you be convergent? Vice versa for your positive terms, they must diverge as well. If both signs of terms converge then how are you not absolutly convergent? So, now we have established that the sum of terms of like signs must diverge.
Now, pick a real number x, any x! (even x factorial!) If x > 0, then you can sum just enough positive terms to so that that sum will be greater than x. Start adding just enough negative terms so that the sum is less than x. You know you need, at least, one negative term to do this. Now, do the same thing you were doing with the positive terms, before. That is, add just enough terms to exceed x. Again, you know you need at least one term to do this. This is very important. Again, sum the negative terms the same way you were, before. Repeat this algorithm so that the sum will waffle between being greater than and less than the target number, x.
Because we know that the absolute value of the terms in the series limits to 0 (or would people like me to prove that?), we know that this order of terms will sum arbitrarily close to the target number. Furthermore, we know that everytime you switch signs you will use, at least, one term of that sign and that you must, repeatedly and endlessly switch signs. Thus, we know that we will use all the terms from both signs and that will make this a true reordering of our series.
Without loss of generality, you can see that this works regardless of what value x is. Thus, a conditionally convergent series can be reordered to sum to any value.
QED

I hope you enjoyed it!