Answer
$2-2x^2+2x^4-2x^6+...+2(-x^{2n})+...$
Work Step by Step
We are given $f(x)=\frac{2}{1+x^2}=\frac{2}{1-(-x^2)}$
for $r$ in $(-1,1)$
The Taylor series for $\frac{1}{1-x}$ is
$1+x+x^2+x^3+...+x^n+...$
The Taylor series for $f(x)=\frac{2}{1-(-x^2)}$ is
$f(x)=2.1+2(-x^2)+2(-x^2)^2+2(-x^2)^3...+2(-x^2)^n+...$
$=2-2x^2+2x^4-2x^6+...+2(-x^{2n})+...$
