I have two strictly increasing integer sequences $a_n$ and $b_n$ such that $\lim_{k\to\infty} \frac{\sum_{n=0}^k a_n}{\sum_{n=0}^k b_n}$ exists.
What can I say about $\lim_{n\to\infty} \frac{a_n}{b_n}$?
Specifically I'd like for these two limits to be equal, but maybe this is asking for too much.
How about the converse: if the limit of ratios exists, then what about the limit of the partial sums?