Sum of the Geometrico-Factorial series

Question

How can I find the sum of the series $$1 + x + 2!\cdot x^2 + 3!\cdot x^3 + \dots + n!\cdot x^n\quad?$$

I was solving this just out of fun but now it doesn't give away. How to form a general formula for such a series? I have been trying my might and even tried wolfram alpha but it answers me in terms of the complex gamma function and exponential integral function $(\operatorname{Ei})$. Is there a simpler formula and if not how can I derive this huge thing?

https://mathworld.wolfram.com/FactorialSums.html

Answer 1

The series $\sum n! x^n$ is divergent when $x\neq 0$.

Littlewood wrote (in the preface for Hardy's book "Divergent series"):

"... Abel wrote in 1828: "Divergent series are the invention of the devil and it is shameful to base on them any demonstration whatsoever". In the ensuing period of critical revision they were simply rejected. Then came a time when it was found that something after all could be done about them ..."

You will find some ideas on the sum of the series $\ \sum n! (-x)^n\ $ in the book "Divergent series" written by Hardy (p 26-29).