[emerald000]

The actual sum does not give -1/12 using normal mathematics. There a lot of videos going on around showing how to get that result using simple algebra on infinite divergent series. That's something you can't do. It's like those proofs of 1 = 2 that divide by 0 to get to their result.

Anyway, there is a branch of mathematics called analytic continuation. Without going into detail, it deals with how analytic functions behave outside of their domain. And there is this well-known function called the Riemann zeta function defined as the sum of 1/n^s for s from 1 to infinity. This function is only defined for s>1. But if we were to take s = -1, we would get 1 + 2 + 3 + .... And using those rules of analytic continuation, you can get a value outside of the range of functions and in this case, the value is -1/12.

So what is the value of the infinite sum? We made a mistake by just asking this question. Divergent sums don't have value until we give them one. The question is not what is the value but what we should define it as? There a couple of ways to go about it and each have their strengths and weaknesses. The usual way involving limits would not give any answer. The analytic continuation (and a couple others) gives -1/12. Since something is more useful that nothing, we decided to give the infinite sum the value -1/12. But neither is more correct than the other.

PS: If you want to read more about it: http://terrytao.wordpress.com/2010/0...-continuation/