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

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