Closed form for a series involving exponential functions

Question

Is there a closed form for the series $\sum_{n=0}^{\infty} \frac{(-1)^n}{n!}\mathrm{e}^{-n \theta}$ ?

The series is convergent for all real values of $\theta$ as $\lim_{n \to \infty} \frac{\mathrm{e}^n}{n!}=0$.

Answer 1

$\sum_{n=0}^{\infty} \frac{(-1)^n}{n!}\mathrm{e}^{-n \theta}=\sum_{n=0}^{\infty} \frac{(-e^{-\theta})^n}{n!}= e^{-e^{- \theta}}$.