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 a_k \le b_k for all k \in \mathbb{N} and \sum_{k=1}^{\infty} b_k is abs. convergent, then \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 \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 0 < \alpha < \infty. If |a_k|^{{\alpha/k}} \to a_0, where a_0 < 1, then \sum_{k=1}^{\infty} a^{\alpha}_k is abs. convergent.