Sum of series ln(1+(1/n))-1/(n+1)

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

True

False