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.