Test for convergence of the series $\sum\limits_{n=3}^\infty\frac{1}{(\ln \ln n)^{\ln n}}$

Question

Test for convergence of the series $\sum\limits_{n=3}^\infty\frac{1}{(\ln \ln n)^{\ln n}}$

I test the series by this steps $(\ln \ln n)^{\ln n} = (e^{\ln \ln \ln n})^{\ln n}= (e^{ \ln n})^{\ln \ln \ln n }=n^{\ln \ln \ln n}$.

Then $(\ln{ {\ln \ln{n}}}) > 2$.

Thus, $\frac{1}{(\ln \ln n)^{\ln n}} < \frac{1}{n^2}$ and thus the series converges by the comparison test. Is this work..!? are there some wrongs ..!?