I have two alternating partial sums $a_n=\sum_{j=0}^n (-1)^j f_j$ and $b_n=\sum_{j=0}^n (-1)^j g_j$, where $f_j$ and $g_j$ are positive numbers. I know that $a_n\to +\infty$ and $b_n\to +\infty$ as $n\to \infty$ and that, in the same limit, $f_n/g_n \to c$, where $c$ is a constant.
How can I study $\lim_{n\to+\infty} {a_n}/{b_n}$? I am aware of the Stolz–Cesàro theorem, but I don't think it can be applied here due to the alternating nature of the partial sums.