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