Let $(a_j), (b_j), (c_j)$ be finite sequences of non-negative real numbers, with $c_j>0$ for all $j$ and decreasing monotonically. To make things more concrete, say that $c_j=e^{-j}$. I'm looking for a way to bound $$\frac{\sum\limits_{j=1}^N a_jc_j}{\sum\limits_{j=1}^N b_jc_j}$$ above by some "good" multiple of $$\frac{\sum\limits_{j=1}^N a_j}{\sum\limits_{j=1}^N b_j}$$ or $$\frac{\sqrt{\sum\limits_{j=1}^N (a_j)^2}}{\sqrt{\sum\limits_{j=1}^N (b_j)^2}}.$$ I say good because, of course, it is bounded above $$\frac{\max c_j}{\min c_j}\frac{\sum\limits_{j=1}^N a_j}{\sum\limits_{j=1}^N b_j}$$ and also by
$$\frac{1}{2}\left(C+\frac{1}{C}\right)\frac{\sqrt{\sum\limits_{j=1}^N (a_j)^2}}{\sqrt{\sum\limits_{j=1}^N (b_j)^2}},\qquad C=\sqrt{\frac{\max b_j\max c_j}{\min b_j \min c_j}},$$
but these are not remotely tight. I am unsure of a way to make use of the monotonicity of $c_j$, which I feel like should help. I'd also be interested, if it is easier, in expressions like $$\frac{\sum\limits_{j=1}^N a_jc_j}{\sum\limits_{j=1}^N b_jc_j}=\frac{\sum\limits_{j=1}^N a_j}{\sum\limits_{j=1}^N b_j}+\mathcal{O}(?),$$ and/or similarly with the $\ell_2$ norms provided that $\mathcal{O}(?)$ is somewhat tight.
EDIT: Simplified problem statement and added a specific common sequence.