Another Laurent Expansion Question

1. The problem statement, all variables and given/known data
Find all possible Laurent expansions centered at 0 for
(z - 1) / (z + 1)

Find the Laurent Expansion centerd at z = -1 that converages at z = 1/2 and determine the largest opens et on which
(z - 1) / (z + 1) converges



2. Relevant equations



3. The attempt at a solution

(z - 1) / (z + 1) breaks down into [z / (z+1)] - [1 / (z+1)]

For the first one divide out the z to obtain 1 / 1 + (1/z) I think? However not being in the form 1 / 1 - (1/z) would this be the same series but negative? That doens't seem right.

For breaking down -1 / (z + 1) I didn't know how to attack that one.
 

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