How to compute this limit? Which could be an example that Cauchy's first theorem doesn't hold for infinite limits.

Question

Compute $\lim\limits_{n\rightarrow \infty} \sum\limits_{k\geq 10}^n\dfrac{(\log(\log k))}{k}$. Actually this example comes to my mind in order to give a counter example that Cauchy's first theorem on limits doesn't hold for divergent sequences, i.e. if $\{a_n\} \rightarrow \infty $ then it is possible that the sequence of averages $\{\dfrac{a_1+\cdots +a_n}{n}\}$ converges.