Taylor series $\log((1-z^3)/(1+z^2))$

Question

I need to find the taylor series of $$\log \frac{1-z^3}{1+z^2}$$ around $0$ and the radius of convergence.

I wrote $\log((1-z^3)/(1+z^2))=\log(1-z^3)-\log(1+z^2)$ And then wrote the Taylor series for each term but I couldn't find a nice way to write this as 1 series.

Any ideas?

Answer 1

Hint:

Use both

$$\frac{1-z^3}{1+z^2}=1-\frac{z^2+z^3}{1+z^2}1-z^2\frac{1+z}{1+z^2}$$

and

$$\ln(1-z)=-\sum_{n=1}^\infty\frac{z^n}n\;,\;\;\text{for}\;\;|z|<1$$

You thus need

$$\left|\frac{z^2+z^3}{1+z^2}\right|<1\iff |z|^2|1+z|<|1+z^2|\implies\ldots etc.$$