It's not a coincidence. I'm saying that both calculations - the math we're using to understand that Casimir effect, and the math we use to 'find' the sum equals -1/12, are making the same simplification that 'misses the point' of what's 'actually happening'. In some sense that's hard to put one's finger on.

You could get any result you want from this kind of proof. Someone else here already used some of these steps to prove that 1+1+1+1+1...==0.