Function expansion in power series

Question

I am trying to expand the function $f: \Bbb C \setminus\{ 1\} \to \Bbb C, f(z):= \frac{1}{1-z}$ into a power series with center c = -1. That seems to me like the sum of a geometric serie for a $ |z| <1$. how should i shift the center? Can anybody help me with this example please. Thank you

Answer 1

Add and subtract $1$ in the denominator $$f(z)=\frac{1}{1-z}=\frac{1}{2-(z+1)}=\frac{1}{2}\frac{1}{1-(z+1)/2}$$

Then use that $\frac{1}{1-x}=1+x+x^2+...$ with $x=(z+1)/2$.