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I struggle with the second line of this...
If Z = 1 - 1 + 1 - 1 + 1 - 1...
1 - Z = 1 - [1 - 1 + 1 - 1 + 1 - 1...]
This leads me to believe 1 - Z =/= Z.
The pattern of Z alternates addition and subtraction, while the addition of the 1 - doubles up on subtraction. If we assume the pattern continues and you stop at a given point, the two results do not match (perhaps someone smarter than I could prove).
After 3...
1 - [1 - 1 + 1] = 0
1 - 1 + 1 = 1
After 4...
1 - [1 - 1 + 1 - 1] = 1
1 - 1 + 1 - 1 = 0
After 5...
1 - [1 - 1 + 1 - 1 + 1] = 0
1 - 1 + 1 - 1 + 1 = 1
If you need a further example of what I am seeing, let's rewrite as
Z = 1 + (-1) + 1 + (-1) + 1...
1 - Z = 1 + (-1) + (-1) + 1 + (-1) + 1...
The alternating nature of the pattern is not the same, we have double (-1), so:
1 - Z =/= Z
This ruins the rest of that series of equations.
I feel like I am missing something, perhaps someone with more experience could explain.