Let x = 1 + 2 + 4 + 8 +16 . . .

Then x-1 = 2 + 4 + 8 + 16 . . .

and (x-1)/2 = 1 + 2 + 4 + 8 . . .

so (x-1)/2 = x

so x - 1 = 2x

so -x = 1

and x = -1

so

1 + 2 + 4 + 8 + 16 . . . = -1