Suppose you add up one plus two plus four plus eight plus sixteen plus… well, you
get the point.
Each number is twice as big as the previous one, so it's pretty obvious that this sum
But wait a second!
We can multiply the whole thing by one and it'll be the same, right?
And one is the same as two minus one… so we have two plus four plus eight plus sixteen
etc., minus one minus two minus four minus eight minus sixteen…
So almost everything on the right side cancels, and voila: 1+2+4+8+16, and so on, is negative