What's bigger, the sum of powers or the power of the sum? Do we know if $(\sum\limits_{i=1}^n a_i)^k \geq \sum\limits_{i=1}^n a_i^k$ for any $k\geq1$?

If the $a_i$ are $\ge 0$, then the inequality is clear, since expanding $\left(\sum_1^n a_i\right)^k$ we get all the terms $a_i^k$, and also "cross-terms."
If some of the $a_i$ are allowed to be negative, let $k=n=2$. Let $a_1=1$ and $a_2=-1$.  Then the left side is $0$ and the right side is $2$, so the inequality need not hold.  
