1 + 2 + 3 + 4 + 5 + 6 + . . . = -1/12Even common sense will correctly tell one that is wrong.
Basically for any series
1 + 2 + 3 + 4 + 5 + . . . + n = n(n+1)/2Solving this problem was what tipped Gauss's teachers that he was good at math. (Not that he was the first to solve it, but that he figured it out at a young age and so got special tutoring in mathematics.)
The limit of this series as n approaches infinity is not going to converge on -1/12 no matter what crazy proof they talk about in the video. One cannot legitimately treat the various series the way that they do in the video.
Only if one converts the series into a function (and they are not really the same thing) could one argue that the resultant quadratic equation could be solved to show that it equals a particular pair of irrational numbers plugged into the formula could come out with an answer of -1/12. Since they are not integers, however, they do not work for the actual series. There is no valid way for anything in the series to equal -1/12.
Math may be hard but it certainly not that hard.