Answer
$2x-6x^2+18x^3-54x^4+...(-6x^{n+1})+...$
Work Step by Step
We are given $f(x)=\frac{2x}{1+3x}=\frac{2x}{1-(-3x)}$
for $r$ in $(-\frac{1}{3},\frac{1}{3})$
The Taylor series for $\frac{1}{1-x}$ is
$1+x+x^2+x^3+...+x^n+...$
The Taylor series for $f(x)=\frac{2x}{1-(-3x)}$ is
$f(x)=2x.1+2x(-3x)+2x(-3x)^2+2x(-3x)^3...+2x(-3x)^n+...$
$=2x-6x^2+18x^3-54x^4+...(-6x^{n+1})+...$
