So each problem is a true or false and should be either proven or a counterexample shown.

1) (I think I understand this one, but I am not super confident) If $\displaystyle a_k \le b_k$ for all $\displaystyle k \in \mathbb{N}$ and $\displaystyle \sum_{k=1}^{\infty} b_k$ is abs. convergent, then $\displaystyle \sum_{k=1}^{\infty} a_k$ converges. I think this is false. If we use a_k = -1 and b_k = 0 we satisfiy our initial conditions but this implies that $\displaystyle \sum_{k=1}^{\infty} a_k$ is divergent.

2) (this one I am lost on completely, I cannot figure out if it's true or false). Suppose $\displaystyle 0 < \alpha < \infty$. If $\displaystyle |a_k|^{{\alpha/k}} \to a_0$, where $\displaystyle a_0 $< 1, then $\displaystyle \sum_{k=1}^{\infty} a^{\alpha}_k$ is abs. convergent.