Show that the series is absolutely convergent The series is
$$\sum^\infty_{n=2} \frac{(-1)^n}{n(\ln(n))^3}$$
I tried the ratio test which did not do anything.  I also tried the root test which gave me
$$\frac{-1}{\sqrt[n]{n}\cdot (\ln(n)^3-n)}$$
which I don't think is right.  Is their another test I can do to confirm that this series is absolutely convergent?  Thanks in advance.

$\textbf{Integral Test}$
$$\sum_{n=2}^{\infty} \frac{1}{n (\ln(n))^3} \ \ \ \text{converges} \iff \lim_{R\to \infty} \int_{2}^{R} \frac{1}{x(\ln(x))^3}\ \text{dx}\ \ \text{converges}$$
Let $u =\ln(x), du = \frac{1}{x} \ \text{dx}\ $ ... (go from here!)
