## Calculus with Applications (10th Edition)

For $r$ in $(-\infty,\infty)$ The Taylor series for $f(x)=e^{-2x^2}$ is $=1-2x^2+2x^4-\frac{4x^6}{3}+...+\frac{(-2x^2)^n}{n!}...$
We are given $f(x)=e^{-2x^2}$ for $r$ in $(-\infty,\infty)$ The Taylor series for $e^x$ is $1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+...+\frac{1}{n!}x^n+...$ The Taylor series for $f(x)=e^{-2x^2}$ is $f(x)=1+(-2x^2)+\frac{(-2x^2)^2}{2!}+\frac{(-2x^2)^3}{3!}+...+\frac{(-2x^2)^n}{n!}+...$ $=1-2x^2+2x^4-\frac{4x^6}{3}+...+\frac{(-2x^2)^n}{n!}...$