Sum of series (2k+1)(2k+2)

Given number:
$$\left(2 k + 1\right) \left(2 k + 2\right)$$
It is a series of species
$$a_{k} \left(c x - x_{0}\right)^{d k}$$
- power series.
The radius of convergence of a power series can be calculated by the formula:
$$R^{d} = \frac{x_{0} + \lim_{k \to \infty} \left|{\frac{a_{k}}{a_{k + 1}}}\right|}{c}$$
In this case
$$a_{k} = \left(2 k + 1\right) \left(2 k + 2\right)$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{k \to \infty}\left(\frac{\left(2 k + 1\right) \left(2 k + 2\right)}{\left(2 k + 3\right) \left(2 k + 4\right)}\right)$$
Let's take the limit
we find

True

False