Brute force: $$\exp{(-x)}=\sum_{n=0}^{\infty}(-1)^n\frac{x^n}{n!}$$

Recursion relation for $$ \exp{(-x)}=\sum_{n=0}^{\infty}s_n=\sum_{n=0}^{\infty}(-1)^n\frac{x^n}{n!} $$ $$ s_n=-s_{n-1}\frac{x}{n}, $$ $$ \exp{(x)}=\sum_{n=0}^{\infty}s_n $$ $$ \exp{(-x)}=\frac{1}{\exp{(x)}} $$