Does the series $\sum_{n=2}^{\infty}\frac{\sqrt{n^2+1}-n}{\ln(n)}$ converge? I need to find if the following series converges absolutely:
$$\sum_{n=2}^{\infty}(-1)^n\frac{\sqrt{n^2+1}-n}{\ln(n)}$$
I know that the series itself is a leibniz series and thus converges, but I don't know if it's converges absolutely.
The "new" series is
$$\sum_{n=2}^{\infty}\bigg|(-1)^n\frac{\sqrt{n^2+1}-n}{\ln(n)}\bigg|=\sum_{n=2}^{\infty}\frac{\sqrt{n^2+1}-n}{\ln(n)}$$.
I tried to use the ratio test, but got $1$, and I can't find a series to compare with it.

$$\frac{\sqrt{n^2+1}-n}{\ln(n)} = \frac{n \left(\sqrt{1 + \frac{1}{n^2}}-1\right)}{\ln(n)} = \frac{ \frac{1}{2n} + o \left( \frac{1}{n^2}\right)}{\ln(n)} \sim \frac{1}{2 n \ln (n)}$$
so your series is not absolutely convergent.
