Answer
The Taylor series for $f(x)=\ln(1+2x^4)$
for $r$ in $(-\infty,\infty)$ is
$=x-2x^8+\frac{8}{3}x^{12}-4x^{16}+...+\frac{(-1)^n(2x^4)^{n+1}}{n+1}+...$
Work Step by Step
We are given $f(x)=\ln(1+2x^4)$
for $r$ in $(-\infty,\infty)$
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 $f(x)=\ln(1+2x^4)$ is
$f(x)=x-\frac{1}{2}(2x^4)^2+\frac{1}{3}(2x^4)^3-\frac{1}{4}(2x^4)^4+....+\frac{(-1)^n(2x^4)^{n+1}}{n+1}+...$
$=x-2x^8+\frac{8}{3}x^{12}-4x^{16}+...+\frac{(-1)^n(2x^4)^{n+1}}{n+1}+...$
