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I'm not sure what you mean by "doubles up on subtraction." it looks like a math error to me. The equation
1 - Z = 1 + (-1) + (-1) + 1 + (-1) + 1...
is wrong. The correct derivation is
Given: Z = 1 - 1 + 1 - 1 + ...
1 - Z = 1 - (1 - 1 + 1 - 1 + ...)
1 - Z = 1 - 1 + 1 - 1 + 1 - ...
1 - Z = Z
You got all of the previous (finite) examples right, but they are irrelevant. The equation 1-Z=Z works only for the INFINITE series.