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

   = 1 + 2(1 + 2 + 4 + 8 + ...)

   = 1 + 2s,

so

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

(Regular summation gives a sum of infinity, but the idea above does make a certain sense.)

(Thanks, Murtadha.)