I suppose you mean
[tex]f(x)=\dfrac{x^3}{x^2+1}[/tex]
Recall that for [tex]|x|<1[/tex], we have
[tex]\dfrac1{1-x}=\displaystyle\sum_{n=0}^\infty x^n[/tex]
Then
[tex]\dfrac1{1+x^2}=\dfrac1{1-(-x^2)}=\displaystyle\sum_{n=0}^\infty(-x^2)^n=\sum_{n=0}^\infty(-1)^nx^{2n}[/tex]
which is valid for [tex]|-x^2|=|x|^2<1[/tex], or more simply [tex]|x|<1[/tex].
Finally,
[tex]f(x)=\displaystyle\frac{x^3}{x^2+1}=\sum_{n=0}^\infty(-1)^nx^{2n+3}[/tex]
