TITLE: Convergence of $\sum _{n=2}^{\infty }\:\frac{1}{n\ln\left(n\right)}$
QUESTION [5 upvotes]: Is this convergent or not?
$$\sum _{n=2}^{\infty }\:\frac{1}{n\ln\left(n\right)}$$
I tried using the ratio test but the limit is giving me 1, which doesn't help me. I don't think I'm supposed to use the integral test, since we haven't studied it.

REPLY [2 votes]: People are mentioning the Cauchy Condensation Test, without saying what it is. Assuming you're not familiar with it, as a lot of people are not:
If $x_n>0$ and $x_{n+1}\le x_n$ then $\sum x_n<\infty$ if and only if $\sum_k2^kx_{2^k}<\infty$. Proving this is a good exercise; as has been pointed out, it shows your series diverges.