Does $\sum_{n=3}^\infty \frac {1}{(\log n)^{\log(\log(n)}}$ converge?

Does the following series converge:

$$\sum_{n=3}^\infty \frac {1}{(\log n)^{\log(\log(n)}}$$

I've tried all test I know... Any ideas ?

Solution 1:

Note that $(\log n)^{-\operatorname{loglog} n}=e^{-(\operatorname{loglog} n)^2}$, since $\log n=e^{\operatorname{loglog} n}$.

For large $n$, we have $(\operatorname{loglog} n)^2\lt \log n$, so for large $n$ the $n$-th term is greater than $\frac{1}{n}$.

The fact that $(\operatorname{loglog} n)^2$ is eventually dominated by $\log n$ is just the familiar fact that $e^x\gt x^2$ for large enough $x$.

Remark: In dealing with convergence of series, it is often better to ask oneself first: How fast are the terms approaching $0$? Looking instead for a test to use tends to distance us from the concrete reality of the series.