Taylor series for $\ln(\frac{1-z^2}{1+z^3})$ Taylor series for $\ln(\frac{1-z^2}{1+z^3})$
I've tried to $$\ln(1-z^2)-\ln(1+z^3)=\sum (-1)^{3n-1}z^{2n}-\sum(-1)^{4n-1}z^{3n}$$
I didn't manage to make it one series any help is good

Starting from @J.G.'s answer, all symplify to
$$\log\left(\frac{1-z^2}{1+z^3}\right)=\sum_{n=2}^\infty \frac{2 \cos \left(n\frac{\pi  }{3}\right)-1}{n} z^n$$
