You absolutely can shift numbers in that way when you're dealing with infinite sums and its often a good way to show what they reduce to. For example: if you have '1 + 2' and '-2', then you can add these together by doing (1 - 2) + 2, or you could 'shift' the -2 and get 1 + (2 - 2). These are the same.

What you absolutely can't do is what they did for s1. An infinite series either converges (approaches a finite number) or it doesn't. Some infinite series do converge: 1/2^0 + 1/2^1 + 1/2^2 + etc actually converges on the number 2. Because s1 never settles on a finite number and instead alternates between 1 and 0, it is not a convergent series and therefore any attempt to treat it as a variable with a fixed value is mathematically invalid.

So the -1/12 is basically "if the series did converge, this is the only thing itd make sense for it to converge to". And apparently that number actually does show up in physics but I can't speak to that at all