convergence and nested logs

Question

The problem is to test convergence for the series: $\sum^\infty_{n=3}1/(\ln n)^{\ln(\ln(n))}$

I tried manipulating the log term (by means of $\ln(n)^{\ln(n)}=e^{\ln(\ln(n))^{\ln(n)}}$=$n^{\ln(lnn)}$)-- but it's not getting me anywhere.

Answer 1

L'Hospitals rule gives us$$\lim_{n\to \infty}\dfrac{(\ln\ln n)^2}{\ln n}=0$$

Hence for large enough $n$, $(\ln\ln n)^2<\ln n$, thus $$e^{(\ln\ln n)^2}<e^{\ln n} \implies e^{(\ln\ln n)(\ln \ln n)}<e^{\ln n}\\\ \implies \ln n^{(\ln \ln n)}<n \implies \frac{1}{(\ln n)^{\ln \ln n}}>\frac{1}{n}$$ hence series diverges by comparison test.