Study the convergence of the $$\sum_{n=1}^\infty \ln^{1/3}\left(n\tan\frac{1}{n}\right)$$
I'm trying to study the convergence of the series and I need a little help. This is what I tried to do. I used the limit comparison test $$\lim_{n\to \infty}\frac{\ln(n\tan\frac{1}{n})}{n\tan\frac{1}{n}-1}=1$$ This implies that $$\sum_{n=1}^\infty\left(\ln(n\tan\frac{1}{n})\right)^\frac{1}{3} \sim \sum_{n=1}^\infty\left(n\tan\frac{1}{n}-1\right)^\frac{1}{3}$$
And this is where I got stuck because I don't know how to study the convergence of the new one. Should I try another method or should I proceed with this one? If so how can I do that?
The answer in the solution says that it diverges