That will give you the Laurent series that converges when |z|<2. In the region |z|>2 you need to write it as
.
In general, the Laurent series for
will consist of powers of
in the region |z|<a, and powers of
in the region |z|>a.
So for the function
there are three separate Laurent expansions, obtained by adding the power series expansions of the partial fractions. In the region |z|<1 you need to use positive powers of z/2 for the first fraction, and positive powers of z for the other two fractions. In the region 1<|z|<2 you need to use positive powers of z/2 for the first fraction, and negative powers of z for the other two fractions. In the region |z|>2 you need to use negative powers of z/2 for the first fraction, and negative powers of z for the other two fractions.