That 1/2 instantly bothered me. Who decides that average of fluctuation between 2 values is their sum? I understand how it works out to call the sum 1/2, but on

That 1/2 instantly bothered me. Who decides that average of fluctuation between 2 values is their sum? I understand how it works out to call the sum 1/2, but only if the definition of "sum" is changed. More accurate description of number would be something like "convergence of average of sums of Nth element, where N is going to infinity" (call it X). And when looked at it like that, we see that other numbers, like 1/4 are understandable too. But the -1/12 is not even X. It is a number that we get by manipulating X of one series with X of a couple of other series by applying some arbitrary rules that doesn't necessarily make the number mean what the mathematician in the video thinks it means. I am sure the number can be made useful in certain problem, but to call it "sum of infinite series" is just plain wrong.