Honestly, I don't buy the proof. When you take a formula that's valid for a certain range of inputs and just use it outside that range anyways, you're no longer in the land of proof. At that point, you're in the land of "if we extend out mathematical system, we get this". And that's fine, but you don't wind up with "sum(1...inf) === -1/12", you wind up with "in this other mathematical system, sum....".

I'm not going to blame you. As the video explain, this is not a straightforward and obvious generalization of summation.

I think that the generalization that use averages are fine, and some analytic continuations using power series are fine. Analytic functions are just too good to ignore them. It is not another mathematical system. It it just the extension of our mathematical system.

Anyway, to get a result for this sum, you must drop too many of the obvious and expected properties of summation. For example if you add a zero in front of the sum, the result changes :(. There was a nice blog post I can't find, but there is a hint of the problem in https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B... . So the result is ... polemic.

Anyway, in some courses of a Math degree, one of the important ideas is that you have many possible definitions of convergence. For example, if you have a sequence of functions, you have https://en.wikipedia.org/wiki/Uniform_absolute-convergence and https://en.wikipedia.org/wiki/Weak_convergence_(Hilbert_spac... and many many many more.

So you don't write

  lim f_n -> f,

you must write

  lim_{something} f_n -> f

where something explain which definition of convenience you are using.

The trick here is to hide the nasty problem of the multiple definitions of convergence in the dots. So the correct statement of the problem is

1+2+3+4+..._{with a weird convergence}=-1/12

but it doesn't look as nice as

1+2+3+4+...=-1/12

Nice answer.