$$ \frac{1}{z-i} = \frac{1}{z}\frac{1}{1-\frac{i}{z}} = \frac{1}{z} \sum_{n=0}^\infty \frac{i^n}{z^n} = \sum_{n=0}^\infty i^n \frac{1}{z^{n+1}} = \sum_{n=1}^\infty i^{n-1} \frac{1}{z^n} $$