Check the convergence and divergence of

Question

Let $u_n$

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

I don't have any idea which test should i apply

Please just give me a hint,further i will try to solve this.

Thankyou

edit:

Answer 1

You can use the comparison test, with the harmonic series. It turns out that$$\lim_{n\to\infty}\frac{\frac1{\log(n)^{\log(\log n)}}}{\frac1n}=\lim_{n\to\infty}\frac n{\log(n)^{\log(\log n)}}=\infty,$$a proof of which can be found here. Therefore, your series diverges.