I have to find all of the Laurent series for the function
$$ f(z)=\frac{z}{(z+1)(z-2)} $$ about $z=0$
I'm a little confused about the regions that I'm dealing with. I started with the partial fraction decomposition: $$ f(z)=\frac{1}{3(z+1)}+\frac{2}{3(z-2)} $$
Coming out from $z=0$ there's a Taylor series up until $z=-1$ that I guess works for both singularities? i.e. for $|z|<1$ the Laurent series is: $$\frac{1}{3}\sum_{n=0}^{\infty}((-1)^n-(\frac{1}{2})^n)z^n$$
Is the above right? If not how do you go about finding the rest of the Laurent series for $f(z)$? What I really need to understand is how you need to look at the regions because I'm finding it really confusing for this problem.
Thank you!