What's the sum of all integers? I remember from somewhere in the distant past of my math education that you can sum up all the integers and get a wide variety of different results. For instance, you can write the sum as S = 0 + (1 + -1) + (2 + -2) + ... where each pair cancels to 0, so S must be 0. But you can also write it as S = 0 + 1 + (2 + -1) + (3 + -2) + ... where each pair adds up to 1, so S must be infinity. With a little more work it seems like you could also get other answers. My question is: are these answers correct, or are all but one of them mistakes like the numerous "proofs" that 3=2? If they're all correct, how does it make sense that the same numbers can add up to different things when you get to infinity? If they're not all correct, what's the fundamental mistake?
posted on Oct 5, 2005 - 29 answers