1 = 0

Proof:
1 = 1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + ....
  = (1 + -1) + (1 + -1) + (1 + -1) + (1 + -1) + ......
  = 0

kekekeke

>>46
Who said anything about it necessarily being an infinite series?

The OP's attempt and 18 both work for any arbitrarily large, finite number of As and Bs. It doesn't even have to be a large number; as long as there's one variable instance, the proof still works.

Nice try on that technicality.