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 $b_n = 2^{-n}$.