Laurent series expansion in annulus with centre in pole

Question

I am having a hard time expanding the function: $$ f(z) = \frac{1}{(z-1)^2(z+1)} $$ in powers of $(z-1)$ in the region $ D : 1 < |z-1| < 2 $
I already rewrote the function as: $$ f(z) = \frac{1}{(z-1)^2(z+1)} = \frac{1}{2(z-1)^2} - \frac{1}{4(z-1)} + \frac{1}{2(z+1)} $$
Now I am not sure how to turn the term $ \frac{1}{(z-1)} $ for example into a geometric series so that it converges for $ 1 < |z-1| $ or $ |z-1| < 2 $. I even found some formulas in our book (the book is in German) but using them I get a 0 in the denominator, as the centre point of the region is also a pole of my function. I would greatly appreciate any help!