Hi there. I have to find the Laurent series for $\displaystyle \frac{1}{z-2},z_0=\infty$

So what I did was calling $\displaystyle z=\frac{1}{\omega}$, I don't know if this is right.

Then:

$\displaystyle f(\omega)=\frac{1}{\frac{1}{\omega}-2}=\frac{-1}{2-\frac{1}{\omega}}=\frac{-1}{2}\frac{1}{1-\frac{1}{2\omega}}=\frac{-1}{2}\sum_0^{\infty}\left ( \frac{1}{2\omega} \right )^n,|\omega|>\frac{1}{2}$

And then back to z:

$\displaystyle f(z)=\frac{-1}{2}\sum_0^{\infty}\left ( \frac{1}{2} \right )^n z^n,|z|<2$

I have to make the series for |z|>2, but I see that it gives the same as when I centered the series in zero, so I think I'm doing something wrong.