Special case in power series

Question

I know how to write $e^x$ as power series. How can I write $e^{-x}$ as power series? Is it $$ \sum_{n=0}^\infty (-1)^n \frac{x^n}{n!}? $$

Answer 1

Since $$ e^x = \sum \frac{ x^n }{n!} $$

then

$$ e^{-x} = \sum \frac{ (-x)^n }{n!} = \sum \frac{ (-1)^n x^n }{n!} $$

Answer 2

Substitute $-x$ for $x$ in the power series for $e^x$: $e^{-x}=\sum_{n=0}^\infty \frac{(-x)^n}{n!}=\sum_{n=0}^\infty\frac{(-1)^nx^n}{n!}$.