In the numberfile video he doesn't explain the way he calculates 2・S_2. He says he shift it, but doesn't explain why that is valid.

    Shifted version:
         1-2+3-4+5-6 ...
           1-2+3-4+5 ...
    sum: 1-1+1-1+1-1 ...
    
    Multiplied version:
         2-4+6-8+10-12 ...

The multiplied version shifts between +(2n) and -(2n). Following the logic that S_1 = 0.5, because that is the average between 0 and 1, I would argue that the multiplied version of S2 should equal 0, as that is the average between a postitive constant and its negative (but the variance is going to be infinite. Doesn't that have a say?).

What if we triple shift?

    Triple shifted version:
         1-2+3-4+5-6+7-8+9 ...
               1-2+3-4+5-6 ...
    sum:     2-3+3-3+3-3+3 ...

Look! now 2・S2 is equal to 2!