What is the power series $($centered about $a=0)$ for the function?

$f(x)=\frac{x}{1+x^2}$

${\displaystyle \underset{n=0}{\overset{}{}}}(-1{)}^n{x}^{2n}=1-x^2+x^4-x^6+$cdots

${\displaystyle \underset{n=1}{\overset{}{}}}{x}^{2n+1}=x^3+x^5+x^7+x^9+$cdots

${\displaystyle \underset{n=0}{\overset{}{}}}(-1{)}^n{x}^{2n+1}=x-x^3+x^5-x^7+$cdots

${\displaystyle \underset{n=1}{\overset{}{}}}{x}^{2n}=x^2+x^4+x^6+x^8+$cdots

${\displaystyle \underset{n=0}{\overset{}{}}}{x}^{2n+1}=x+x^3+x^5+x^7+$cdots