# Determine convergence of the series $\sum_{n=1}^\infty\frac{1}{\ln(n)^{\ln(n)}}$

How to determine convergence of the series: $$\sum_{n=1}^\infty\frac{1}{\ln(n)^{\ln(n)}}$$

I spent most of the time using the Integral criteria (since the function $f(x)=\frac{1}{\ln(x)^{\ln(x)}}$ is non increasing positive function) but didn't manage to get a solution. I used partial integration method, however it kept getting more and more complex.

Is there any better way of determining convergence of the series?

• Hint: For every $n$ large enough, $$(\ln n)^{\ln n}\geqslant n^2.$$
– Did
Jun 21 '15 at 22:02
• By the way, $\frac{1}{\ln n^{\ln n}}$ is $\frac{1}{0^0}$. While I'm of the opinion that $0^0$ should be defined as $1$, enough people disagree. Jun 22 '15 at 0:04

It converges. $$\log(x)^{\log x} = x^{\log\log x}$$ and for every $x$ big enough we have $\log\log x\geq 2$.