1 + 2 + 3 + 4 + ... = -1/12
If Trump wins many more primaries he may end up with less than zero delegates!
That's wrong.
The "average of the two" (0 and 1 = ½, for S1) is wrong, you cannot do so because the function is not continuous. (The rule they're using only works for continuous functions.)
Secondly, the S2 is not proven, both because it relies on the fraudulent S1 and because the "shifted form" is not proven to be equal to the actual series (1-2+3-4+5-6...) as it could be noted that the negative n is one less than the next n, this would make the series 1+1+1+1+1...
Thirdly, the S that is "solved for" depends on both S1 & S2 which, as we've seen, cannot be concluded are as asserted.