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You're not justified in shifting the terms over. You can't arbitrarily add 0-values to an infinite sum. And if you think about it conceptually you cannot move a line that extends to infinity, you would be stretching/compressing it.
Also to compare the functions S and S' vertically by adding 0 at the beginning of S', then you would need to add 0 at the end of S. This is obviously not possible.