Convergence of $\sum_{n=3}^{\infty} \frac{(\log(\log(n))^2}{n(\log(n))^2}$.

Question

I need to show that the series $$\sum_{n=3}^{\infty} \frac{(\log(\log(n))^2}{n(\log(n))^2}$$ converges.

I'm trying to find some upper bound for it, since the tests i've used so far did not lead to any useful conclusion. I think that a good upper bound might be the series $\sum_{n=3}^{\infty} \frac{1}{n(\log(n))^{1+\epsilon}}$, for $\epsilon >0$ because then i could use the fact that the series $\sum_{n=3}^{\infty} \frac{1}{n^{1+\epsilon}}$ converges. Therefore i tried to approximate the expression $(\log \log n)^2$ to something like

$$(\log \log n)^2 \le (\log n)^{1+\delta}$$

for some $\delta > 0$ but without success. Could you please help me?

Answer 1

Hint: for $n$ big enough, $$ (\log \log n)^2 < \sqrt{\log n } $$ Now use the fact that $$ \sum \frac 1{n(\log n)^{3/2}}<\infty $$