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