I'm trying to find the series of $$f(x)=\sqrt{1-x^{3}}$$ Can I just use the fact that $$\frac{1}{1-x}=\sum x^{n},\quad|x|<1$$ writting $x^{3}$ in the place of $x$ and then, getting this: $$\begin{align*} \frac{1}{1-x}=\sum x^{n} & \Rightarrow \frac{1}{1-x^{3}}=\sum x^{3n} \\ & \Rightarrow \left(\frac{1}{1-x^{3}}\right)^{-1/2}=\left(\sum x^{3n}\right)^{-1/2} \\ & \Rightarrow \sqrt{1-x^{3}}=\sum\frac{1}{\sqrt{x^{3n}}} \end{align*}$$
I don't know, it looks wrong to me. I know I can multiply or divide by anything different from $n$, but is it right?
Thanks :)
Hint: Let $z:=-x^3$, and try to use binomial series for $\sqrt{1+z}$ then write back $z=-x^3$.