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.
 A: 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.
A: You are correct that the alternating series gives you the answer you want about convergence (although not about absolute convergence).
In general, these tests are phrased as: "If such-and-such a condition is true, then the series converges." If you like, think of this in terms of predicates: you have a statement of the form $P \implies Q$. However, given that, $\sim\! P$ (the negation of $P$) tells you nothing at all.
Note that if the condition fails, then this simply gives us no information. It could converge or diverge; in fact your example here is a perfect example---you know the series converges (by the alternating series test), but the ratio test tells you nothing at all.
