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Infinity or -1/12?

You can't add two infinite series the way you did.

[ Z+Z ≠ 1-1+1-1+1-....+1-1+1-1+1-… .]

However,

Z+Z = 2-2+2-2+2-… ≈ 1,

which is consistent with, Z≈ 1/2.

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