Suppose the series $\sum_{k=1}^\infty x_k = +\infty$ diverges where $x_k > 0$ for all $k$.
Does there always a positive sequence $\{y_k\}_{k=1}^\infty$ where $\displaystyle \sum_{k=1}^\infty \frac{1}{y_k} = +\infty$ and $\displaystyle \frac{x_k}{x_{k+1}} < \frac{y_{k+1}}{y_k}$ for all $k$?
I found examples like $\displaystyle x_k = \frac{1}{\sqrt{k}}$ and $\displaystyle \frac{1}{y_k} = \frac{1}{k}$ where it's true but can't come up with a proof that it is always true.
Take $\displaystyle y_k = \frac{\sum_{j=1}^k x_j}{x_k}$ and show that $\sum y_k^{-1}$ diverges.
Since $S_k = \sum_{j=1}^k x_j$ is increasing, we have
$$\sum_{k=n+1}^m y_k^{-1} = \sum_{k=n+1}^m \frac{x_k}{S_k} > \frac{S_m - S_n}{S_m} = 1 - \frac{S_n}{S_m}.$$
Since $S_m \to \infty$, the RHS is greater than $1/2$ for a sufficiently large $m$ greater than any chosen $n$. Thus, by violation of the Cauchy criterion, $\sum y_k^{-1}$ diverges.
Note that $x_ky_k = \sum_{j=1}^k x_j$ is increasing, so $x_k/x_{k+1} < y_{k+1}/y_k$ for all $k \in \mathbb{N}$.