MacLaurin series of $(1-x^2) /(1+x^3)$

Question

I wonder what is the generator term (an) for the MacLaurin series of $(1-x^2) /(1+x^3)$.

Thanks, Amin

Answer 1

We may use the geometric series:

$$\begin{align}\frac{1-x^2}{1+x^3}&=\frac1{1-(-x^3)}-\frac{x^2}{1-(-x^3)}\\&=\sum_{n=0}^\infty(-x^3)^n-x^2\sum_{k=0}^\infty(-x^3)^k\end{align}$$