It's not a very good video. Those kinds of vid do often put things in a rather sensational manner, but in this case they've stepped too far into the realm of the inaccurate and the misleading in order to be more exciting.
It's worth noting that the guy is a physicist, not a mathematician, and so it's possible he doesn't fully understand the material. When he says "the sum really is 1/12", what he really means is that assigning it a value of 1/12 gives us physical predictions which are actually correct, which is indeed interesting.
But he shouldn't just call it a "sum" without any explanation. What he's doing is not really a sum, it's just like a sum in some ways.
If you add up the first term (1), and then the first two terms (1 + 2 = 3), and then the first three terms (1 + 2 + 3 = 6), and so on, you get a sequence of numbers which just get bigger and bigger. Therefore the sum does not exist. This is in contrast with a series like 1/2 + 1/4 + 1/8 + ... , where adding up the successive numbers gets you closer and closer to 1, and we say the sum is 1.
Note that summing is therefore a way of assigning a number, which we call "the sum", to some strings of numbers.
Now, what the guy is really talking about is a different way of assigning a number to the string of numbers "1, 2, 3, ... ". In this sense the "sum" is just a function from strings to numbers, which doesn't pose us any troubles. We are free to define arbitrary functions. For example, I can define a function F, which sends "1, 2, 3, ... " to 100, "1/2, 1/4, 1/8, ... " to pi, and every other infinite sequence to 1. Of course, this F is completely arbitrary and useless, and has no correspondence to "summing". But there are other functions which will share some properties of classical summing. One important property we should require is that, for any string which does have a classical sum, like "1/2, 1/4, 1/8, ... ", the function will send it to the correct number (in this case, 1). Another property might be that, if the function assigns the number A to "a1, a2, a3, ... ", and the number B to "b1, b2, b3, ... ", then it will assign the number A + B to "a1 + b1, a2 + b2, a3 + b3, ... ". There are a bunch of functions with these properties, which also assign values to strings like "1, 2, 3, ... "  and the value turns out to be 1/12. We then might call such a function a "summation method", but that's just a name. The important point is that these functions turn out to be welldefined, interesting, and even physically useful.


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