zeta of -1 is -1/12 directly, without any infinities.
Anyway, it's like how for z in (-1,1) you have
[math] \dfrac {1} {1-z} = 1 +z+ z^2 + z^3 + ... [/math]
and if you plug in
z = d-1
where
d = 1/2^999999,
i.e. some unimaginably small number
then
[math] \dfrac {1} {2-d} = 1 + (d - 1) + (d-1)^2 + (d-1)^3 + ... [/math]
you also see the
1-1+1-1+1-1+... = 1/2
result.
If the sum pops up in physics as some inherently simple finite objectQ, but e.g. z=-1 is the edge of a wall and the way the theory is written down doesn't want to deal with it, then people would also say Q at the wall is 1/2, and not incoorporate analytic limits in their language
Is it literally true for sums as defined in analysis? No, only the limit of the sum.
But if you come up with a thoery of physics, the heuristic version of your theory might not allow for the fancy overhead and so whenever you talk to people who aren't experts in theoretical physics and math, you'd go on about arguing why those sums end up this and thay number in some sloppy way. The complex analysis language is also not integral part of it, merely it's tools are used when necessary.
A tiny minority of physicists has ab idea about rebormalization in quantum field theory.
