Is the series $\sum_{n=2}^\infty \frac{1}{(\log (n))^{\log (\log (n))}}$ convergent or divergent?

Question

Is the series

$$\sum_{n=2}^\infty \frac{1}{(\log (n))^{\log (\log (n))}}$$

convergent?

I know that $ \log(\log n) >2$. Now $\log(\log(\log (n))) > \log 2$. After that I am not able to proceed further.

Answer 1

Note that since $x\ge \log^2(x)$ for $x\ge \log(2)$, we have

$$\begin{align} \int_2^L \frac{1}{(\log(x))^{\log(\log(x))}}\,dx&\overbrace{=}^{x\mapsto e^x}\int_{\log(2)}^{\log(L)}\frac{e^x}{x^{\log(x)}}\,dx\\\\ &=\int_{\log(2)}^{\log(L)}e^{x-\log^2(x)}\,dx\\\\ &\ge \log(L/2) \end{align}$$

Hence, the integral test guarantees that the series of interest diverges.

Answer 2

HINT

Let use the Cauchy condensation test

$$ 0 \ \leq\ \sum_{n=1}^{\infty} f(n)\ \leq\ \sum_{n=0}^{\infty} 2^{n}f(2^{n})\ \leq\ 2\sum_{n=1}^{\infty} f(n)$$

that is

$$\sum_{n=2}^\infty \frac{1}{(\log n)^{\log\log n}} \ge \frac12\sum_{n=2}^\infty \frac{2^n}{(\log 2^n)^{\log\log 2^n}}=\frac12\sum_{n=2}^\infty \frac{2^n}{(n\log 2)^{\log (n\log 2)}}$$

Answer 3

Hint: $a^b$ $= (e^{\log a})^b$ $= e^{b\log a}$. Applying this to your denominator turns it into $e^{(\log\log n)^2}$, and now you should be able to compare with $e^{\log n}$...