I'm sorry, but why did you substitute Z with zero? Where the hell did you get that? I think you forgot that the equation is not a function, and you can't substitute it with a value of zero, since we don't know the value of Z yet. You can't contradict a statement and substitute it with a value since the person is still trying to find the value. Also, 1-1+1-1+1... is Grandi's series, search it, and its been proven that the sum IS negative one, as proven by the diverging averages of the partial sums to one-half
(1/1), (1+0)/2, (1+0+1)/3, (1+0+1+0)/4, ... (1+0+1+0+1+0+1+0+1+0+1+0+1+0+1)/15, and so on. As you can see, it slowly alternates between other numbers and 1/2, but the difference between the alternates diverges to zero as the number of partial sums increase, thus, the equation does diverge to one-half.