Question: Find the power series about the origin for the given function: $\frac{z^{3}}{1-z^{3}}\text{ , }|z|<1$
Answer:
$\begin{align*}
\frac{z^{3}}{1-z^{3}}&=\frac{z^{3}+1-1}{1-z^{3}}\\
&=\frac{z^{3}-1}{1-z^{3}}+\frac{1}{1-z^{3}}\\
&=-1+\frac{1}{1-z^{3}}\\
&=-1+\sum_{n=0}^{\infty} (z^{3})^{n}\\
&=-1+1+\sum_{n=1}^{\infty} z^{3n}\\
&=\sum_{n=1}^{\infty} z^{3n}\text{ , where }|z|<1\\
\end{align*}$
