Oh, one of the neater mathematical 'paradoxes' I've seen is finding the sum of a divergent series (obviously you can't treat infinity this way ... but it's a fu

Oh, one of the neater mathematical 'paradoxes' I've seen is finding the sum of a divergent series (obviously you can't treat infinity this way ... but it's a fun trick). (excuse the notation, I'm doing this with plain ASCII characters ...): Let the sum of the geometric series 1 + 2 + 4 + 8 + 16 + ... be called (=) S. Now multiply both sides by 2: 2S = 2 + 4 + 8 + 16 + 32 + .... Now subtract the second equation from the first: S - 2S = (1 + 2 + 4 + 16 + ...) - (2 + 4 + 8 + 16 + 32 + ...) Notice that all the terms except for the '1' cancel out on the right: -S = 1 And simplify: S = -1 Therefore: 1 + 2 + 4 + 8 + 16 + ... = -1 ;-)