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.

Hint: Absolute convergence of
$$
\sum_{n=1}^\infty a_n
$$
is equivalent to convergence of
$$
\sum_{n=1}^\infty|a_n|
$$
Thus, absolute convergence of the series given is equivalent to convergence of
$$
\sum_{n=1}^\infty\frac1{n\log(n)^3}
$$
A good test to try when logs are involved is the Cauchy Condensation Test. This would mean testing whether
$$
\sum_{n=1}^\infty2^n\frac1{2^n(n\log(2))^3}
=\frac1{\log(2)^3}\sum_{n=1}^\infty\frac1{n^3}
$$
converges.
