Difference between finite partial sums from two divergent series

Question

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 $\left\vert\, \sum_{N\leq i \leq M} (r_i-s_i) \,\right\vert < \dfrac{\epsilon}{2}$? 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 also small enough to be close to $\sum_{N\leq i \leq M}r_i$.

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 $\sum_{N_2\leq i \leq M_2}s_i\in\left( \dfrac{\epsilon}{2}, \epsilon \right)$, I do not know how to find relations between those two integer intervals. Any hints will be appreciated.

Answer 1

A necessary and sufficient condition for $\{s_i\}_{i\in\mathbb N}\subset(0,1)$ to satisfy your property $(*)$, as it is, is simply that $$|s_i-r_i|\le\frac{r_i}2\, \text{ for all } i\in\mathbb N.$$

Indeed, if $\{s_i\}_{i\in\mathbb N}\subset(0,1)$ satisfies $(*)$, then for every $i\in\mathbb N$ and for every $\epsilon$ in the non-empty interval $(r_i,2r_i\wedge1)\subset(0,1)$, taking in particular $N=M=i$, since $\sum_{j=N}^Mr_j=r_i\in (\dfrac\epsilon2,\epsilon)$, we have by $(*)$ $|s_i-r_i|=\sum_{j=N}^M|s_j-r_j|<\frac{\epsilon}2$, therefore also $|s_i-r_i|\le\frac{r_i}2$.

Conversely if $|s_i-r_i|\le\frac{r_i}2$ for all $i\in\mathbb N$ then for every $\epsilon\in (0, 1)$, for every $N, M\in\mathbb{N}$ with $N\leq M$, whenever $\displaystyle\sum_{N\leq i \leq M}r_i \in \left( \dfrac{\epsilon}{2}, \epsilon \right)$, we have $$\left\vert\, \sum_{N\leq i \leq M} (r_i-s_i) \,\right\vert\le \sum_{N\leq i \leq M} |r_i-s_i|\le$$ $$\le \frac12 \sum_{N\leq i \leq M} r_i<\frac\epsilon2,$$ so property $(*)$ holds.