I didn't watch the numberphile video and just read this.

Can someone explain to me why I am allowed to add a padding zero at the start without taking it out at the end?

By adding it, I am giving the sum an offset, and I can kind of understand why adding zero is not a big deal, but they reasoned:

Then S1 + S1 = (1 - 1 + 1 - 1 ...) + (1 - 1 + 1 - 1 ...)

= (1 - 1 + 1 - 1 ...) + (0 + 1 - 1 + 1 - 1 ...)

= (1 + 0) + (1 - 1) + (1 - 1) ....

= 1 + 0 + 0 ... = 1

But I might as well leave the leading Zero out and argue that:

Then S1 + S1 = (1 - 1 + 1 - 1 ...) + (1 - 1 + 1 - 1 ...)

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

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

= (2 - 2) + (2 - 2) + (2 - 2)... = 0