I am trying to study the convergence of the series:  $$\sum_{n =2}^{\infty} \frac{1}{(\ln n)^{\ln n}}$$ First I have failed to use the D'Alembert test  $$ \lim_{n\to \infty} \frac{(\ln (n+1))^{\ln (n+1)}}{(\ln n)^{\ln n}} =1.$$ So I can Conclude anything from here.  I have also tried the integral Comparaison test but it was not soo clear.
Could I have a hint for testing the convergence of the following series please?  $$\sum_{n=2}^\infty\frac{1}{(\ln n)^{\ln n}}$$  I am very appreciative for your help.
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