real analysis - Studying the character of $sum_{n=3}^infty frac{1}{n(log(log n))^{alpha}}$

I have to study the character of this series
$$\sum_{n=3}^\infty \frac{1}{n(\log(\log n))^{\alpha}}$$

with $\alpha$ a real parameter.

Considering the Cauchy condensation test, the equivalent series is:
$$\sum_{n=1}^\infty 2^n\frac{1}{2^n[\log(\log 2^n)]^{\alpha}}=\sum_{n=1}^\infty \frac{1}{[\log(n\log 2)]^{\alpha}}$$

Using the ratio test:

$$\lim_{n\rightarrow \infty} \frac{[\log(n\log 2)]^{\alpha}}{[\log((n+1)\log 2)]^{\alpha}}=\lim_{n\rightarrow \infty} \frac{1}{[\log(\log 2)]^{\alpha}}= \frac{1}{[\log(\log 2)]^{\alpha}}\sim \frac{1}{(-0,36)^{\alpha}}$$

Then when $(-0,36)^{\alpha}>1$ the given series converges, otherwise it diverges

if $\alpha =1$ ,$-0,36<1$ , diverges

if $\alpha =0$ ,$1<1$ , diverges

if $\alpha =-1$ ,$(-0,36)^{-1}=-2,77<1$ , converges

if $\alpha >1$ ,$(-0,36)^{\alpha}<1$ , converges

if $\alpha <-1$ ,$(-0,36)^{\alpha}<1$ , converges

if $|\alpha| <1 , \ne 0$ sometimes it doen't exist $(-0,36)^{\alpha}$
Can someone help me?

Answer

HINT

Let use limit comparison test with

$$\sum \frac{1}{n}$$