Sum of series sqrt(n/(2n+2))

Given number:
$$\sqrt{\frac{n}{2 n + 2}}$$
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
$$a_{n} = \sqrt{\frac{n}{2 n + 2}}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty}\left(\frac{\sqrt{n} \sqrt{2 n + 4}}{\sqrt{n + 1} \sqrt{2 n + 2}}\right)$$
Let's take the limit
we find

True

False