-1/12 is the result of applying zeta function regularization or Ramanujan summation to the sum of the positive integers. It's arguably interesting, but hardly amazing to the vast majority of people who have never heard of those techniques. But the thing that really annoys me is all the people presenting it as the finite limit of a divergent series (this is the default meaning of "=" after an infinite series, if you're using a non-standard meaning you have to specify that!). The first of those videos does this! It's nonsense, and this kind of sloppy approach only encourages contempt for mathematics.