The Bell numbers satisfy $\frac{\ln B_n}{n} \sim \ln n$ which is faster than exponential, so the ordinary generating function $\sum B_n x^n$ has zero radius of convergence. As a more elementary argument, <a href="https://en.wikipedia.org/wiki/Dobi%C5%84ski%27s_formula">Dobinski's formula</a>

$$B_n = \frac{1}{e} \sum_{k \ge 0} \frac{k^n}{k!}$$

establishes that $B_n$ grows at least as fast as $k^n$ for any positive integer $k$, which also implies that $B_n$ grows faster than exponentially and so $\sum B_n x^n$ has zero radius of convergence.

This does not imply it is nonsense to study this series; <a href="https://en.wikipedia.org/wiki/Formal_power_series">formal power series</a> can be studied abstractly and it's common practice to do so in combinatorics. $x$ is just never specialized to a concrete value in $\mathbb{C}$. As another example, $\sum n! x^n$ makes sense as a formal power series also despite having zero radius of convergence, and there are various interesting things to say about it, e.g. its logarithm 

$$\log \sum n! x^n = x + \frac{3}{2} x^2 + \frac{13}{3} x^3 + \frac{71}{4} x^4 + \dots $$

counts <a href="https://qchu.wordpress.com/2015/11/03/the-answer-to-the-puzzle/">subgroups of the free group $F_2$</a>.