In reply to ablackett:
Here's a way to look at it: the Riemann zeta function
http://en.wikipedia.org/wiki/Riemann_zeta_function assigns to every s the number given by adding together 1/n^s for all positive integers n.
s --> Sum_{n=1}^{infty} n^{-s}
So evaluated at s=2, you get the sum over all inverse squares, which is (pi^2)/6. Evaluated at s=3, you get the sum of all inverse cubes, and so on. At s=1 you get the sum of all inverse integers, the harmonic series, which diverges to +infinity, so the Riemann zeta function is infinity at s=1.
This actually defines a well defined complex number, for any complex number s with real part at least 1, so the Riemann zeta function is well-defined on the part of the complex plane right of the line Re(s)=1. (See Wiki for pictures).
Now the weird part. It turns out that the values of the Reimann zeta function can be unambiguous and smoothly defined for other complex numbers that have real part one or less -- this is a process called analytic continuation. This defines the Riemann zeta function on the whole complex plane (give or take the odd point like s=1 where it blows up to infinity), and it turns out that at s=-1 it takes the value -1/12.
Now plug s=-1 into the original definition of the Riemann zeta function (forgetting for a moment that it only really makes sense for Re(s)>1). You get the sum of all the positive integers! So there is a sense in which the sum of all the positive integers is "equal" to -1/12.
This might sound like nonsense, but it actually has applications in physics. In quantum field theory, infinities are frequently "regularized" away -- and one method is called zeta function renormalization. It involves replacing a sum of all positive integers (clearly divergent) by the Riemann zeta function value of -1/12.
Two incredible things then happen. Firstly, when physicists do these ridiculous renormalization schemes, they work. That is, they can predict correctly with extraordinary accuracy the behaviour of quantum fields (I don't know if zeta function renormalization itself has been used in practise though). Secondly, occassionally mathematicians are able to make rigorous the physicists' techniques, and when they do, they find that the correct value was -1/12 all along (this certainly happens in the conformal field theory of the free boson, for example). So Hardy and Ramanujan's intuition, crazy though it seems, was, as usual, bob on.