Sum of series 1/(3n+2)(3n-1)

Given number:
$$\frac{3 n - 1}{3 n + 2}$$
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
$$a_{n} = \frac{3 n - 1}{3 n + 2}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\left(3 n + 5\right) \left|{3 n - 1}\right|}{\left(3 n + 2\right)^{2}}\right)$$
Let's take the limit
we find

True

False