Rob, you remembered right, as ThomasC's post explains in the spoiler.
Here's a fallacy that illustrates a different concept. Can you spot the error in this "proof" that 1 = 0? The ... in each line here means to repeat the pattern on into infinity.
Nope, that wasn't the intention. The formatting here isn't ideal, but look at the difference between lines 2 and 3, the intent there is that I've just re-arranged the parentheses from line 2, all of them being shifted one spot to the left. I'll change it to make the # of 1's match up, now there are 7 of them in both line 2 and in line 3:
But that was what I was getting at. In line 2, it's of the form 1 + n(1-1), where n can be infinite. In line 3, it's just n(1-1), which isn't the same thing.
You're getting close...that property does apply to certain kinds of infinite sums, but not to others. Full explanation soon, unless someone tries to kill me first.
Using the associative property (i.e., moving parentheses around without changing the value of anything) for infinite sums only works if the series, when written without any parentheses, converges to a finite number (and in a specific way, to boot, convergence alone isn't always enough, but I'll leave that alone for now). For instance, the infinite sum:
1/2 + 1/4 + 1/8 + 1/16 + 1/32 + ...
...equals 1 when summed to infinity. You can re-arrange the terms any way you like, group them in any way with any pattern of parentheses, and the infinite sum will still converge to the value of 1. But let's see what the infinite sum from my previous posts looks like without any parentheses:
1 - 1 + 1 - 1 + 1 - 1 + 1 - ....
And so forth. Does this infinite sum have a finite value? No, it diverges. Certain sums are obviously divergent because they keep getting bigger in an unstoppable way, i.e. 1 + 2 + 3 + 4 + ..., or 1 + 1 + 1 + 1 + ..., for instance. The series in question here doesn't do that, as each successive 1 is cancelled out by the next -1, but it diverges because it never approaches one particular value in an inexorable way. If you stop after finitely many steps, you get a sum of either 1 or 0, depending on where you stop.
So you can't re-arrange any parentheses in this case because the infinite sum does not actually represent a finite value in and of itself.