We'd like to make sense of the sum

1+2+3+4+...

This appears to make no sense, but it starts to make sense if you throw in the right exponent. If s is a real number greater than 1, then the sum

1-s+2-s+3-s+4- s+...

makes perfect sense. For example, if s=2, this sequence converges to π2/6 for reasons you can read about here. In fact, you can take s to be any complex number with real part greater than 1, and this sequence will still converge. In that case, we abbreviate our infinite sum as ζ(s). Now ζ(s) is a differentiable function at those values where it's defined. The sum we're interested in is ζ(-1), which is not (yet) defined. But it turns out that there is exactly one way to extend the function ζ so that it can be applied to any non-zero complex number and still remain differentiable. That one and only possible extension gives ζ(-1)=-1/12. All of this turns out to be incredibly important for reasons that are too deep for this short note. But it is the real reason why 1+2+3+4+... ``ought'' to equal -1/12---just as Euler predicted.