It looks as if there is a lot of fluff in these infinite series and somehow when we remove the redundancy and noise, we get the gold nugget which is -1/12 in this case. The surprising part is that such series are a common occurrence in physics and quantum theories (string) and we get the right results (observable and measurable) by replacing such series with -1/12.
Similar to Schrodinger’s wave function which gives out the right results each and every time but could not explain why it is THE function and why it exists, similar to Ramanujam’s mock theta functions which he just wrote from his dreams but could not recollect how he arrived there, in the same manner the above substitution works but we are currently unable to explain in the realm of the theories (number theory) we are using to pose the question and answer it. May be it is the case that we are seeing only a limited view point of a larger dimensional space, a larger context. This is however consistent with the work of Godel where there exist true statements that cannot be proved within the system (cannot be both consistent and complete at the same time). One has to include external axiom(s) into the system to be able to prove the working result. Just like sqrt(-1) cannot be solved within the real number system, however has solutions within the complex number system.
But for now -1/12 appears to be the finite part of this infinite lump.