Find the Taylor series of $f(z)=\frac{z}{i-3z^2}$ at $z_0=0$

Question

Find the Taylor series of $f(z)=\frac{z}{i-3z^2}$ at $z_0=0$. Is there a simple way to find the expression of $f^{(n)}(0)$?

$$f(z)=\sum_{n=0}^\infty \frac{f^{(n)}(z_0)}{n!} (z-z_0)^n= \sum_{n=0}^\infty \frac{f^{(n)}(0)}{n!} z^n.$$

Answer 1

$f(z)=-iz \frac 1 {1+3iz^{2}}=-i \sum_{n=0}^{\infty} z(-3iz^{2})^{n}$ valid for $|z| <\frac 1 {\sqrt 3}$.