#Laurent Expansion of 1/((2-z)(1-z)) for |z|>2

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tropic rover
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Hi lovely people, I can't for the life of me figure out where the first term '1/z' comes from in the last Laurent Series in the Picture...
Using Laurent's theorem I do get a residue of 1 indeed but algebraically I am very confused to say the least.
The second photo shows my calculation which does not yield a '1/z' term.

I would love some help or a useful hint. Thank you 🙂