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.