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This is a follow up on this question. It asks whether following equality holds $$\lim\limits_{N\to\infty}\frac{\sum\limits_{i=1}^N a_i}{\sum\limits_{i=1}^N b_i} = \lim\limits_{N\to\infty}\frac{1}{N}\sum_{i=1}^N\frac{a_i}{b_i}$$

for $a_i>0$, $b_i>0$, $a_i<b_i$.

It was established that it does not with counterexample. If we take $a_i=\frac{1}{3^i}$ and $b_i=\frac{1}{2^i}$ then left hand side evaluates 0.5 and right hand side 0.

Simple simulation shows convergence. See following plot of left hand side minus right hand side. It is with $a_i$, $b_i$ uniformly distributed in range $(0,1)$, such that $a_i<b_i$.

simulation

Nonetheless, this is no proof.

Question: what would be the conditions under which this equality holds?

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https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_de_Stolz-Ces%C3%A0ro

Applying this result in your case will give you a sufficient condition like :

$$\lim_{n\to\infty}\frac{a_n}{b_n} \to l\quad \text{ and }\quad\sum_{i=1}^n b_i \to_{n\to\infty}\infty $$

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