Expand the function \begin{equation*} f(z)=\frac{z}{1+z^{3}} \end{equation*} a) in a series of positive powers and b) in a series of negative powers. In each case, specify the region in which the expansion is valid.
I feel like I'm missing something obvious here because I don't even know where to start i.e. I don't know what annulus to consider for this problem, or how to apply the formula given: \begin{equation*} a_{n}=\frac{1}{2\pi i}\int _{C_{R}} \frac{f(\zeta)}{\zeta^{n+1}} d\zeta \end{equation*} if $n\geq 0$ and \begin{equation*} a_{n}=\frac{1}{2\pi i}\int _{C_{r}} \frac{f(\zeta)}{\zeta^{n+1}} d\zeta \end{equation*} if $n< 0$. What would my $C_{R}$ and $C_{r}$ be in this case? Any help would be appreciated and I'm sorry if my wording is confusing. Thanks so much!
If $\lvert z\rvert<1$, then $|z^3|=|z|^3<1$ and then\begin{align}\frac z{1+z^3}&=z(1-z^3+z^6-z^9+\cdots)\\&=z-z^4+z^7-z^{10}+\cdots\end{align}And, if $\lvert z\rvert>1$, then\begin{align}\frac z{1+z^3}&=-z\left(\frac1{z^3}-\frac1{z^6}+\frac1{z^9}-\cdots\right)\\&=-\frac1{z^2}+\frac1{z^5}-\frac1{z^8}+\cdots\end{align}