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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. –  André Nicolas Jan 27 '12 at 17:59
    
you cant say anything about $\lim a_n/b_n$ –  yoyo Jan 27 '12 at 18:36
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1 Answer 1

To make things simpler, let $r_n = \frac{a_n}{b_n}$. All that we know about $r_n$ is that it is always a rational number. In particular, note that it need not be strictly increasing, because $a_n$ and $b_n$ need not increase at the same rate. For example $a = 1, 8, 16, 128$ and $b = 2, 4, 32, 64...$ so that $r_n = \frac{1}{2}, \frac{2}{1}, \frac{1}{2}, \frac{2}{1},...$

If $r_n$ is oscillating like this, the limit as $n \to \infty$ of $r_n$ won't converge.

However, $$\sum_{n=0}^k a_n = (1 + 8) \cdot 16^0 + (1 + 8) \cdot 16^1 + \ldots (1 + 8)\cdot 16^{k/2}$$ and $$\sum_{n=0}^k b_n = (2 + 4) \cdot 16^0 + (2 + 4) \cdot 16^1 + \ldots (2 + 4)\cdot 16^{k/2}$$

Therefore,

$$\lim_{k \to \infty} \frac{\sum_{n=0}^k a_n}{\sum_{n=0}^k b_n} = \frac{(1+8)}{(2 + 4)}\cdot \frac{\sum_{n=0}^k 16^{k/2}}{\sum_{n=0}^k 16^{k/2}} = \frac{9}{6}$$

and the limit of the ratio of the sums converges, but does not imply that the limit of the ratio converges.

On the other hand, it is possible that both limits exist. For example, if $a_n = n$ and $b_n = 2^{-n}$.

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