Sum of series ln(n/10)/(exp(n/10)-1)

Given number:
$$\frac{\log{\left(\frac{n}{10} \right)}}{e^{\frac{n}{10}} - 1}$$
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
$$a_{n} = \frac{\log{\left(\frac{n}{10} \right)}}{e^{\frac{n}{10}} - 1}$$
and
$$x_{0} = 0$$
,
$$d = 0$$
,
$$c = 1$$
then
$$1 = \lim_{n \to \infty} \left|{\frac{\left(e^{\frac{n}{10} + \frac{1}{10}} - 1\right) \log{\left(\frac{n}{10} \right)}}{\left(e^{\frac{n}{10}} - 1\right) \log{\left(\frac{n}{10} + \frac{1}{10} \right)}}}\right|$$
Let's take the limit
we find

False

False

False