Answer
The Taylor series for $f(x)=\ln(1-5x^2)$
for $r$ in $(-\frac{\sqrt 5}{5},\frac{\sqrt 5}{5})$ is
$x-\frac{25}{2}x^4-\frac{125}{3}x^{6}-\frac{625}{4}x^{8}+...+\frac{(-1)^n(-5x^2)^{n+1}}{n+1}+...$
Work Step by Step
We are given $f(x)=\ln(1-5x^2)=\ln[1+(-5x^2)]$
for $r$ in $(-\frac{\sqrt 5}{5},\frac{\sqrt 5}{5})$
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 $f(x)=\ln(1-5x^2)$ is
$f(x)=x-\frac{1}{2}(-5x^2)^2+\frac{1}{3}(-5x^2)^3-\frac{1}{4}(-5x^2)^4+....+\frac{(-1)^n(-5x^2)^{n+1}}{n+1}+...$
$=x-\frac{25}{2}x^4-\frac{125}{3}x^{6}-\frac{625}{4}x^{8}+...+\frac{(-1)^n(-5x^2)^{n+1}}{n+1}+...$
