Let $\lim \_{n\to\infty}a\_n=l$. Show that $\sum\_{n=1}^\infty \frac{1}{n^{a\_n}}$ converges if $l>1$ and diverges if $l<1$. What happens if $l=1$?

I tried to use the ratio test, but could not get a good estimate, I have difficulties, that there is the series $a\_n$ involved. I know that $\sum\_{n=1}^\infty \frac{1}{n^p}$ converges if $p>1$ and diverges if $p\geq 1$ but I am not sure how to use this exactly. 

For $l=1$, I guess both things could happen? Definitely, we can take $a\_n=1$ for all $n$ and then we get the harmonic series which is divergent. Is there an example where the series converges?