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Difference between finite partial sums from two divergent series

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Fix a sequence $(r_i)_{i\in\mathbb{N}} \subseteq (0, 1)$ such that $\lim_i r_i=0$ and $\sum_{i\in \mathbb{N}} r_i=\infty$. According to the answer in this post, for any $c>0$ there exists $N,M\in\mathbb{N}$ with $N\leq M$ such that $\sum_{N\leq i \leq M}r_i \in \left( \dfrac{c}{2}, c\right)$. Now let $(s_i)_{i\in \mathbb{N}} \subseteq (0, 1)$ be another sequence such that $\lim_i s_i=0$ and $\sum_{i\in \mathbb{N}}s_i=\infty$.

My question is the following: is there a necessary condition for the sequence $(s_n)_{n\in \mathbb{N}}$ such that for any $\epsilon\in (0, 1)$, whenever $N, M\in\mathbb{N}$ with $N\leq M$ satisfies $\sum_{N\leq i \leq M}r_i \in \left( \dfrac{\epsilon}{2}, \epsilon \right)$, we will have $\sum_{N\leq i \leq M} s_i\in (\epsilon, 2\epsilon)$? Namely, I wonder when given a fixed range between $N, M$ such that $\sum_{N\leq i \leq M}r_i$ is arbitrarily small, when we can have that $\sum_{N\leq i \leq M}s_i$ is arbitrarily close to $\sum_{N\leq i \leq M}s_i$.

Under our assumption, we always have $\lim_n s_n-r_n=0$. I suppose it is not possible to have the finite partial sum of $(r_n)_{n\in \mathbb{N}}$ be always arbitrarily close to the partial sum of $(s_n)_{n\in \mathbb{N}}$, since both partial sums could be arbitrarily large. Then I am tempted to believe it may be possible if one of, or both partial sums are small. However, given $N_1, N_2, M_1, M_2\in\mathbb{N}$ such that $\sum_{N_1\leq i \leq M_1} r_i\in \left( \dfrac{\epsilon}{2}, \epsilon \right)$ and $\sum_{N_2\leq i \leq M_2}s_i\in (\epsilon,2\epsilon)$, I do not know how to find relations between those two integer intervals. Any hints will be appreciated.


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