investigate convergence using comparison test.

Question

Use the comparison test to investigate the convergence/divergence of the series $$\sum_{n=3}^\infty\frac{1}{(\ln n)^{\ln n}}$$

So i know that i can change $(\ln n)^{\ln n} = n^{\ln \ln n}.$ But how do i continue from here?

Answer 1

hint: $\ln n^{\ln n}= e^{\ln (\ln n)\cdot \ln n}> e^{2\ln n}=n^2$

Answer 2

It is sufficient to find $n$ such that $\ln\ln n>1$. Since $\ln\ln x$ is a monotone increasing function, then for $n>e^{e^2}$ we would have $$\frac{1}{(\ln n) ^{\ln n}}<\frac{1}{n^2}$$