Show that each of the following series converges

1) $\displaystyle \sum\_{k=1}^\infty\dfrac{\log k}{k^p}$, $p > 1$. 

2) $\displaystyle \sum\_{k=1}^\infty \dfrac{1}{k^{\log k}}$.

Can anyone please help me start this problems.
I think I can use the comparison theorem. Thank you.