Limit of $ \sum_{n=0}^{+\infty} \frac{(-x)^n}{1+n!} $ as $x\rightarrow +\infty$.

Question

The series of functions $$ \sum_{n=0}^{+\infty} \frac{(-x)^n}{1+n!} $$ is pointwise convergent for any $x\in \mathbb{R}$, thus it defines the function $f:\mathbb{R}\rightarrow \mathbb{R}$ given by $f(x) = \sum_{n=0}^{+\infty} \frac{(-x)^n}{1+n!}$. Is there a way to evaluate the limit $\lim_{x\rightarrow +\infty} f(x)$?

Answer 1

Write $$ f(z) = \sum_{n=0}^\infty \frac{z^n}{1+n!} $$ of course an entire function. We want to know about $\lim_{z \to -\infty} f(z)$. Maple produced this graph numerically.

So I guess that $\lim_{z \to -\infty} f(z) = +\infty$.