[Math] Taylor Series expansion of $f(z)=(1+z)/(1-z)$

Question

I am trying to determine the Taylor Series expansion of $f(z)=\frac{(1+z)}{(1-z)}$ centered at $z_0=i$ by trying to rewrite it in terms of a geometric series. However, I wasn't able to proceed using this idea. What other methods could be applied in order to solve this problem?

Answer 1

Hint: for the series of $f(z)$ centred at $z=c$, it's convenient to substitute $z = c+t$ so that you're taking the series of $f(c+t)$ centred at $t=0$. Also notice that $$ \frac{1+c+t}{1-(c+t)} = -1 + \frac{2}{1-(c+t)} = -1 + \frac{2}{1-c}\left(\frac{1}{1-t/(1-c)}\right)$$