I originally wrote this as a comment on a bookmark, but thought I might as well stick it here as well.
I really liked maths at school. I also really liked playing rugby at school. At university I realised that there was no way I could continue to play rugby as by that age the level of play was above my physical abilities - basically I would have been snapped in two; I still don't understand why I'm so slight, perhaps I'm lacking in the gym and/or steroids? Sadly the same has kind of happened with maths. At school I could understand it all, well all that I saw - I was unaware I was being purposefully restrained from seeing really advanced maths and was only being treated to the beginnings of calculus, etc. Now it's easy to find and look at really complex mathematics, but it's way above my cognitive abilities.
Having said all that though, I think this article could have been explained more simply; Although maybe not, because Mark CC is more clever than I am so chances are I'm wrong. The problem, for lay persons (me and probably you), when they see:
1+2+3+4+5+… = -1/12
is the equals sign. We think of it in terms of "meaning the same value". That is, we read it in our minds as:
"Adding the series together of one, two and so on, all the way up to infinity, gives us -1/12".
Which is nonsense.
But that isn't what the equals sign means here. From the [Wikipedia page on Mathematical equality](https://en.wikipedia.org/wiki/Equality_(mathematics)), the use of the equals sign here is "that the expressions represent the same mathematical object". Instead we should read in our heads:
"Adding the series of one, two and so on, together all the way up to infinity can be represented by a constant, -1/12"
And that sits much more easily on the mind.