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It converges for |z|>1 (root test or ratio test).
So you know for |z|>1
z/(z-1)=1/(1-1/z)=1+1/z+1/z^2+1/z^3+..
and the convergence is uniform (M test or whatever),
so differentiate both sides to obtain
D(z/(z-1))=-1/z^2-2/z^3-3/z^4-..
So your series is
-zD(z/z-1)=z/(z-1)^2
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