real analysis - Series $sum^{infty}_{k=1}a_k$ and $sum^{infty}_{k=1}b_k$ converge but $lim_{n rightarrow infty}frac{b_k}{a_k}$ diverges

EDIT: The are many good, simple examples to try here, but I want to understand this "hint," given below, opaque as it is.

I'm working on an existence proof that shows two series $\sum^{\infty}_{k=1}a_k$ and $\sum^{\infty}_{k=1}b_k$ converge but $\lim_{n \rightarrow \infty}\frac{b_k}{a_k}$ diverges.

So I need $\forall M \in \mathbb{R}$ with $M >0$ $\exists$ $N \in \mathbb{N}$ such that $\forall k \gt N$ we have $\frac{b_k}{a_k} \gt M$.

My hint is to construct an increasing sequence $(N_j)$ such that $\sum^{\infty}_{k=N_j}a_k \leq \frac{1}{j^3}$ and to let $b_k = ja_k$ for $Nj \leq k \leq N_{j+1}$.

Good. Then the sequence (a_k) can be broken down into blocks, with $\{a_1 + a_2 + ... + a_{N_1-1}\}$ less than some M, and the subsequent blocks less than $\frac{1}{j^3}$. So we have a way to estimate the sum: $\sum^{\infty}_{k=1}a_k \leq M + \sum^{\infty}_{j+1}\frac{1}{j^3}$.

I'm befuddled because I can't figure out how this information about the sums relates to the convergence of $\frac{b_k}{a_k}$ unless there's some test that I'm not thinking of. I thought to construct sequences out of the partial sums, but that seems counter-indicated by the phrasing of the prompt.

Guidance appreciated.