If $a_n\rightarrow\infty$ then $\sum_{i=1}^n\frac{1}{(a_n)^{a_n}}$ converges?

Question

I am trying to prove or disprove the following statement:

If $a_n\rightarrow\infty$ then $\sum_{i=1}^n\frac{1}{(a_n)^{a_n}}$ converges.

I have been trying to find a counter example by testing some small growing sequences such as $a_n=\log\log(n)$, but i could not show the resulting series diverges.

My second try was trying to show there exist some $N>0$ such that ${(a_n)^{a_n}}>n^2$ for all $n>N$ but i guess thats not true anyway.

Any hints will be appreciated.

Answer 1

Here's an example:
$$a_n=\sqrt {\log n}$$.

For $n>1$

We get $a_n^{a_n}<n$ since $a_n\log a_n<(a_n)^2=\log n$.