Given number:
$$\frac{\left(-3\right)^{n + 1}}{2^{3 n}}$$
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
$$a_{n} = \left(-3\right)^{n + 1}$$
and
$$x_{0} = -2$$
,
$$d = -3$$
,
$$c = 0$$
then
$$\frac{1}{R^{3}} = \tilde{\infty} \left(-2 + \lim_{n \to \infty}\left(3^{- n - 2} \cdot 3^{n + 1}\right)\right)$$
Let's take the limitwe find
False
False
$$R = 0$$