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.
I have to solve: $\lim\limits_{x\to \infty} \frac{\ln(x+1)}{\ln(x)}$. Can you give me any hints to go? Thanks a lot!