Suppose I have the limit

$\lim_{m\rightarrow \infty}\frac{\sum_{k=0}^ma_{k,m}}{\sum_{k=0}^mb_{k,m}}$.

When can I write this as

$\lim_{n\rightarrow \infty}\lim_{m\rightarrow \infty} \frac{\sum_{k=0}^ma_{k,n}}{\sum_{k=0}^mb_{k,n}}$?

To be specific, both sums converge to exponentials, which tend to zero as $n\rightarrow$. I'd like to take their ratio before letting $n$ tend to infinity.