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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?

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    The limit of $a_n/b_n$ need not exist.2012-01-27
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    you cant say anything about $\lim a_n/b_n$2012-01-27

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