Complex Analysis: Laurent Series Expansion in region for 22 C.

I have solved the others but I am really struggling on 22c. I need it to converge for $|z|>2$. This is the part I am really struggling with. I am trying to get both fractions into a geometric series with $1/(1-(1/(z/2))$ so that it converges for $1/(z/2)<1$; which becomes, ($z>2$). I however cannot manipulate either series into this form that I am looking for. I have tried many times. I have also tried combining both fractions into one fraction over $z^2-1$, but am not getting anywhere. Any pointers, or tips so that I can get it to converge for $|z|>2$ by manipulating the fractions (just simply cannot get in right form). Thank you.Image url is attached.

Laurent Series

Answer

Note that we can write

$$\begin{align} \frac{1}{z-1}-\frac{1}{z+1}&=\frac{2}{z^2}\left(\frac{1}{1-1/z^2}\right)\\\\ &=\frac{2}{z^2}\sum_{n=0}^\infty \frac{1}{z^{2n}}\\\\ &=\sum_{n=1}^\infty\frac{2}{z^{2n}} \end{align}$$