I am confused about what you are doing here. has a pole at z= 2, as you say, but is analytic at z= 0. A function's "Laurent series" about any point at which it is analytic is just its Taylor series.
In this case, which is the sum of a geometric series with first term a= 1/2 and common ratio r= z/2.
As for "at z=infinity", the Laurent series of any function, f(z), about is just the Laurent series of f(1/z) about z= 0.
If then .