The equation $$ 1 + 2 + 3 + \dots = -1/12 $$ is quite famous.
From the point of view of mathematics, I have no problem with it. My (probably naive) understanding is that there are certain "sums'' which can be assigned to an infinite series through zeta function regularisation, other types of regularisation, Ramanujan summation, and probably quite a few other methods I've never heard of. These "sums" have a lot of nice properties; in particular they agree with the usual sum when both are defined, and so we use the same notation as for the usual sums.
What I would like to understand is how this sum is useful in physics. Now, I have heard in somewhat vague terms that it's useful in string theory and in computation of Casimir force, and presumably in a couple of different places. To the extent that I understand the explanations, they are saying something along the lines:
A certain value with a physical meaning has, according to our computations, the value $\infty - 1/12$. However, we know that infinities do not physically exists, hence the $\infty$ we got has to cancel out with another $\infty$ (coming from some other consideration; something to do with symmetry?). Hence, what we are really left with is $-1/12$.
Is the above in any way an accurate understanding of how this works? If yes, then I have two questions.
Why are we so quick to rule out the infinite value? For instance, if I were to imagine Earth as an infinite half-space with uniform density, then I could compute that the gravity on the surface is infinite (if I recall correctly). The conclusion is that this was not a good model (even though Earth looks quite flat and infinite from where I stand). Why cannot the conclusion in Casimir effect be: Hmm, we have all these contributions which together give a divergent series, we must have forgotten that there's a cutoff, or something.
How do we know that the $\infty$ will cancel out so nicely? In other words, suppose I know that the answer to my problem is a difference between two divergent series, and one of them has the associated "sum" equal to $-1/12$. How do I know that the other one will automatically have the "sum" equal to $0$?