# co.combinatorics – Does the ordinary generating function of Bell numbers converge?

I am working in a field not really based on combinatorics, therefore I appologize if my question is in any kind invalid. Nevertheless, in my calculations, the Bell numbers appeared. I need to find some $$x$$ such that the ordinary generating function
$$B(x) = sum_{n=0}^{infty}B_n x^n$$
converge. I haven’t found the answer nowhere in the literature. On the opposite, there are quite a lot of results concerning $$B(x)$$, but none of them is questioning for which $$x$$ it has some sense. It is evident that the case $$x>1$$ lead to a divergent series, which is not much of an interest. But what about $$x<1$$? I suppose there must be such $$x$$, otherwise it is nonsense to study such series, is it not?

One other thing suprised me. There is a nice representation in Klazar of $$B(x)$$ such that
$$B(x) = sum_{n=0}^{infty}frac{x^n}{(1-x)(1-2x)cdots(1-n x)}.$$
But what if $$x$$ equals to $$1/k$$ for some $$k$$ natural? Would that make the series divergent? I am sorry, but is has been bugging me for some time that there is no explanation in the literature that I have been looking into. Does anyone have some relevant source of information?

Thank you, I would appreciate any help.