Answer
$5x-15x^2+45x^3-135x^4+...+5x(-3x)^n+...$
Work Step by Step
We are given $f(x)=\frac{8x}{1+3x}=\frac{8x}{1-(-3x)}$
with $c=8x$ for $r$ in $\frac{-1}{3}\lt x \lt \frac{1}{3}$
The Taylor series for $f(x)=\frac{8x}{1+3x}$ is
$f(x)=5x+5x(-3x)+5x(-3x)^{2}+5x(-3x)^3+...+5x(-3x)^n+...$
$=5x-15x^2+45x^3-135x^4+...+5x(-3x)^n+...$
