I am trying to express a function with Taylor and Laurent series. I've been reading my textbook and also various online resources, but I still can't follow any of the example problems. Here's what I understand so far.
I have a function of a complex-valued number z, and its denominator is $0$ at $z= 1$ and $2$.
From this, I think I should have three separate series. One for $|z|<1$, one for $1<|z|<2$, and one for $|z|>2$. I think I managed to get the first one by deriving the Taylor series for $f$, but I know that this only has a radius of convergence of $1$.
How can I proceed to derive the other two series? One resource I read stated that when $z$ is greater than the radius of convergence, I can use the fact that $1/z<R$ to work with a geometric series, but I am confused on the algebraic manipulation.
Any feedback/constructive criticism is appreciated.