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.