Warning, prolix rant ahead.

I actually think that video is awful and it certainly doesn't illustrate what Terry is illustrating in his blog post. Rather, the video goes through a bunch of non-rigorous symbolic manipulations which ends with the author writing down the sequence of symbols "1 + 2 + 3 + ... = -1/12".

However, unless one says precisely what one means by "+", "...", and "=" — which Terry does — then we have no way of really saying whether the steps taken to reach the so-called "conclusion" are valid or not. What's more, just because the same sequence of symbols appear in both the video and Terry's blog post doesn't mean they represent the same thing or have anything to do with each other at all.

Put another way, if you and I reach the same conclusion, but you do so rigorously and I do so speciously, that doesn't mean I've proven the same thing as you have.

For example, in the video, why are we allowed to add together two infinite series term by term in the way they describe? It seems "natural," I know, but if that's natural, why can't we also, say, group the addition differently or rearrange the terms? After all, a + (b + c) = (a + b) + c and a + b = b + a, no matter what a and b are. Why can't we write

  S = 1 - 1 + 1 - 1 + ...

as

  S = (1 - 1) + (1 - 1) + ...

If we permit ourselves to do that, well, suddenly the sum "appears" to be 0 and not 1/2. BTW, if you want a somewhat-more rigorous reason for why the sum "should be" 1/2...

      S = 1 - 1 + 1 - 1 + ...

  So then
      
  1 - S = 1 - (1 - 1 + 1 - 1 + ...)
        = 1 - 1 + 1 - 1 + ...
        = S

  which implies S = 1/2

We're not proving that S = 1/2 here, though. We're proving this statement: if it makes sense to talk about S and we're permitted to do the things we just did to S then S = 1/2.

Terry knows all this, of course, which is why he says, "If one formally applies (1) at these values of {s}..." That word "formally" is key here. To a mathematician "formally" means "in a purely symbolic manner without considering whether there's a sensible or consistent way of interpreting these symbols."

So, Terry is saying, "If we treat these sums as purely symbolic entities then when we substitute in s = -1 we get a the purely symbolic statement 1 + 2 + 3 + ... = -1/12." He then goes on to illustrate ways we might make sense of this purely symbolic (formal) sum.

The video, however, is no "proof" of anything at all. It's just a shell game with symbols on a page, relying on people's vague intuition about what we're allowed to do with numbers. Just because the symbols in the video include those we typically take to represent numbers and addition doesn't mean they actually do.