I think, rearranging (1-1+1-1+1-1...) as (1+1+1+1...) - (1+1+1+1...) is not valid because of the "shifting" of values, and decomposing of one infinity into two. If you insist on "proving" equality to 0 - there is an easier way: just use parenthesis like this: (1-1)+(1-1)+(1-1)... = 0 + 0 + 0 ... which is "clearly" zero. Right? Not really.
This has been bugging me all day, and the best "intuitive" explanation may be based in physics: If you draw (1-1+1-1+1-1...) on a graph assuming it's some physical value over time - it's easy to see that 1/2 is the center of oscillation. So, even though the graph never converges to 1/2 - in the infinity it may as well converge. Given the example with a light bulb.. which is ON or OFF, in the infinity, the bulb would be neither ON or OFF - it would be half-bright. If you start sequence with -1, you get an oscillating line around -1/2. So, that makes sense too. If you start doing tricks like (1-1)+(1-1)+(1-1)... = 0 + 0 + 0 ... -- it's easy to see that the trick here is selectively collapsing time intervals to 0, which doesn't make sense in the physical sense.
I started today thinking that (1-1+1-1+1-1...) = 1/2 was a fallacy, but now I think it's actually true, and it starts to make sense. However, I'm still to make the leap to understanding how (1+2+3+4+5+...) = -1/12 can be a useful fact, even if it's mathematically correct.