What is amusing is that Ramanujan said this (that 1+2+3+... = -1/12) in a letter to find a job in England, just to illustrate that he was not bad in computing. He was of course considered as crackpot until the letter was given to Hardy, who recognized immediately the genius. Another amazing thing is that in string theory, the computation of the mass of the photon leads to the expression 1/12 + (1+2+3+4+5+ ...). the physicists thought "shit, again an infinity", but were comforted by the number theorists, that on the complex numbers, this gives "somehow" the needed -1/12.

Number theory is full of that kind of very surprising facts. of course the limit is not the usual limit of a series. We don't get that "infinity" = -1/12.


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