Assignment: Let $\sum_{k=1}^{\infty} a_k$ and $\sum_{k=1}^{\infty} b_k$ be series with positive, real entries. Suppose that the following limit exists:
$q=\lim\limits_{k\to\infty}\frac{b_k}{a_k}$ with $0<q<\infty$.
I'm asked to prove that if $\sum_{k=1}^{\infty} a_k$ is convergent, then so is $\sum_{k=1}^{\infty} b_k$, and vice versa for divergence.
Frankly, I'm at a loss here. I know that the limit $\lim\limits_{k\to\infty} a_k=a$ exists, and that the sequence $(a_k)^{\infty}_{k=1}$ must go to zero, but I haven't a clue how to proceed.
Any advice/pointers would be much appreciated.