suppose that $f(z) = \frac{1}{z(z^2-1)}$. Singularity exist at $0, +1, -1$. There are three largest annuli centering at 1 $\{0<|z-1|<1\}, \{1<|z-1|<2\}, \{2<|z-1|<\infty\}$.
How do you find the laurent series?
What is the intuition that differentiates the formulation of one laurent expansion from the other?
How are these three laurent series different?