The Taylor series expansion for e^x is :
\[e^x = \sum_{n=0}^\infty \frac{x^n}{n!}\]
\[Substitute −x\ into\ the\ Taylor\ series\ for\ e^x \]
\[e^{-x} = \sum_{n=0}^\infty \frac{(-x)^n}{n!}\]
\[Let\ {(-x)^n}\ as\ {(-1)^n}*x^n\]
\[the\ Taylor\ series\ expansion\ for\ the\ function\ f(x) = e^{−x}:\]
\[e^{-x} = \sum_{n=0}^\infty \frac{(-1)^n*x^n}{n!}\]