Answer
$7x-14x^2+28x^3-56x^4+...+7x(-2x)^n+...$
Work Step by Step
We are given $f(x)=\frac{7x}{1+2x}=\frac{7x}{1-(-2x)}$
with $c=7x$ for $r$ in $\frac{-1}{2}\lt x \lt \frac{1}{2}$
The Taylor series for $f(x)=\frac{7x}{1+2x}$ is
$f(x)=7x+7x(-2x)+7x(-2x)^{2}+7x(-2x)^3+...+7x(-2x)^n+...$
$=7x-14x^2+28x^3-56x^4+...+7x(-2x)^n+...$
