Answer
For $r$ in $(-\infty,\infty)$, the Taylor series for $f(x)=x^6e^{-x}$ is
$x^6-x^7-\frac{1}{2}x^8-\frac{1}{6}x^9...+\frac{(-x)^{n+6}}{n!}+...$
Work Step by Step
We are given $f(x)=x^6e^{-x}$
for $r$ in $(-\infty,\infty)$
The Taylor series for $e^x$ is
$1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+...+\frac{1}{n!}x^n+...$
With $c=x^6$, the Taylor series for $f(x)=x^6e^{-x}$ is
$x^6+x^6.(-x)+x^6\frac{(-x)^2}{2!}+x^6\frac{(-x)^3}{3!}+...+x^6\frac{(-x)^n}{n!}+...$
$=x^6-x^7-\frac{1}{2}x^8-\frac{1}{6}x^9...+\frac{(-x)^{n+6}}{n!}+...$
