Answer
$\frac{-x}{2}-\frac{x^2}{8}-\frac{x^3}{24}....+\frac{(-1)^n(\frac{-x}{2})^{n+1}}{n+1}+...$
Work Step by Step
We are given $f(x)=\ln(1-\frac{x}{2})=\ln(1+(\frac{-x}{2}))$
for $r$ in $[2,2)$
The Taylor series for $f(x)=\ln(1+4x)$ is
$f(x)=\frac{-x}{2}-\frac{1}{2}(\frac{-x}{2})^2+\frac{1}{3}(\frac{-x}{2})^3-\frac{1}{4}(\frac{-x}{2})^4+...+\frac{(-1)^n(\frac{-x}{2})^{n+1}}{n+1}+...$
$=\frac{-x}{2}-\frac{x^2}{8}-\frac{x^3}{24}....+\frac{(-1)^n(\frac{-x}{2})^{n+1}}{n+1}+...$
