calculus - Is the series $ L = sum_{i=2}^{infty } frac{1}{n log(n)} $ convergent or divergent?

Does $ L = \sum_{i=2}^{\infty } \frac{1}{n \log(n)} $ converge or diverge?

I established that:
$$
L \le I

= \int^{\infty}_{2} \frac{1}{n \log(n)}
= \lim_{n \to \infty} [ \ln(\log(n)) - \ln(\log(2)) ],
$$
and as $ \lim_{n \to \infty} \log(x) = \infty $, then $ L $ diverges.

But I'm not sure:

1. of the sense of the inequality,


2. about the conclusion.