Wait what? (1+2+3+4+...

[Froge]

Shifting the second S2 to the right so that adding them together recreates S1 is not valid. Let's say S2 grows very large (i.e. it approaches infinity). The fact that the second S2 has been "shifted" to the right implies that it is always one term ahead of the first S2, and this term cannot be neglected even as the sums approach infinity. This can be best illustrated by pretending there is a stop point. The magnitude of the last term of the second S2 is tremendous as S2 approaches infinity so it makes a big difference.

To show an example, let's pretend S2 stops at 10^99:

S2a = 1 - 2 + 3 - 4 + ... + (10^99 - 1) - 10^99
+ S2b = 0 + 1 - 2 + 3 - .... - (10^99 - 2) + (10^99 - 1) - 10^99
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1 - 1 + 1 - 1 + ... + 1 - 1 - 10^99 = -10^99

Notice the -10^99 that wasn't added? I know the "well it should equal S1 when it reaches infinity" but that argument doesn't apply to a series of natural numbers, because S2b will always contain one more term than S2a. Even if the number of terms in the first S2 approach infinity, the number of terms in the second S2 approach a sort of "infinity plus one" (i.e. still one more term than the first S2), which of course makes a giant difference in the sum.

I also think the very first sum S1 isn't correct. 1 - 1 + 1 - 1 + 1 - 1 + ... does not equal 1/2. Not only that, but I don't think the guy doing the proof in the video knows his terminology. He says 1/2 is the "natural number" that we attach to the sum S1, but 1/2 is definitely not a natural number. Of course this is more of an ad hominem rebuttal, but I think it decreases his credibility.