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With my little knowledge on maths I think you made a mistake.I was actually bugged about infinity sums for a while but I slowly start to understand them thankst your maths.You said Z+Z=1-1+1-1...1-1+1-1 which equals to Z.Well there's a problem in your maths!You seem to think that Z is actually ending and another Z begins!The Z function never ends and that's why you can't add something on the end of an infinite function!That's why I think (with my basic math knowledge) you shouldn't add a Z function to the end of another Z function!Because you don't know if the Z function will end with a 1 or a -1!The best approach of how much Z+Z is this:
Z+Z=1+1-1-1+1+1-1-1...=2(1-1+1-1+....)=2*1/2=1
Thank for a great night read :)