Not sure how to go about finding the representative function given a power series:
The one I have is $$\sum_{n=1}^\infty z^{3n}$$
I know $\sum_{n=0}^\infty z^{n}$ is just $\frac{1}{1-z}$ so perhaps this one is just $\frac{1}{1-z^3}$ but I am mostly guessing and would like guidance on how to do it in general
Since$$\lvert z\rvert<1\implies\sum_{n=0}^\infty z^n=\frac1{1-z},$$and since $\lvert z\rvert<1\iff\lvert z^3\rvert<1$ then, if $\lvert z\rvert<1$,\begin{align}\sum_{n=1}^\infty z^{3n}&=\left(\sum_{n=0}^\infty(z^3)^n\right)-1\\&=\frac1{1-z^3}-1\\&=\frac{z^3}{1-z^3}.\end{align}