Comparing asymptotics of various functions Is the first function big oh, theta or omega of the second function in the following examples.  

$(\log n)^{\log n} $ and $ \frac{n}{\log n}$
I am not sure how fast the first function grows.  

$n2^n$ and $3^n$
I think the second function grows faster but not sure how to prove it.  

$(\log n)^{\log n} $ and $2^{(\log_2n)^2}$
This is just puzzling.

$\sum_{i=1 }^n i^k$ and $n^{k+1}$.
And so is this.

Any help is much appreciated.
For the first and the third, taking the logarithm of both sides will really help. For the second, try to use the fact that if $\lim_{n\to\infty} \frac{f(n)}{g(n)} = 0$ then $f(n) = o(g(n))$. For the fourth, try approximating the sum using an integral.