Given number:
$$\frac{1}{2^{x} x!}$$
It is a series of species
$$a_{x} \left(c x - x_{0}\right)^{d x}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{x \to \infty} \left|{\frac{a_{x}}{a_{x + 1}}}\right|}{c}$$
In this case
$$a_{x} = \frac{1}{x!}$$
and
$$x_{0} = -2$$
,
$$d = -1$$
,
$$c = 0$$
then
$$\frac{1}{R} = \tilde{\infty} \left(-2 + \lim_{x \to \infty} \left|{\frac{\left(x + 1\right)!}{x!}}\right|\right)$$
Let's take the limitwe find
$$\frac{1}{R} = \infty$$
$$R = 0$$