sequences and series - Confused about using alternating test, ratio test, and root test (please help).

So I have to determine if $\sum_2^{\infty} \frac{(-1)^n}{ln(n)}$ absolutely converges, conditionally converges, or diverges.

So first I tried the Alternating Series Test, because that is what you do first (right???). (Also I just started understanding mathjax more, so excuse the formatting).

Alternating Series Test

lim
n-> infinity ($\frac{1}{ln(n)}$) = 0

and it's decreasing as well, so that means its convergent.

One question I have here is if one of these attribute of the
alternating series test fails, does that mean it's divergent or I just can't use the test?

Ratio Test

Now to find if it's absolute convergence or conditional convergence, I did the ratio test, but I got 1. That means I can't use this test.

Root Test

I don't see how I can use the root test here because I just raised everything to the 1/n power and I'm stuck.

What should I do?

I hope someone can clear up my confusions. Thanks.

Answer

It converges conditionally and not absolutely because $\displaystyle \sum_{n=2}^\infty \left|(-1)^n\cdot \dfrac{1}{\ln n}\right| = \displaystyle \sum_{n=2}^\infty \dfrac{1}{\ln n} \geq \displaystyle \sum_{n=2}^\infty \dfrac{1}{n} = \infty$, and you've done the first part that series is convergent by the alternating series test.