Given number:
$$\frac{9^{n} n!}{n^{2 n}}$$
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
$$a_{n} = n^{- 2 n} n!$$
and
$$x_{0} = -9$$
,
$$d = 1$$
,
$$c = 0$$
then
$$R = \tilde{\infty} \left(-9 + \lim_{n \to \infty}\left(n^{- 2 n} \left(n + 1\right)^{2 n + 2} \left|{\frac{n!}{\left(n + 1\right)!}}\right|\right)\right)$$
Let's take the limitwe find
$$R = \infty$$