I'm aware of the fact that for two infinite series $\sum_{n=1}^\infty a_n$ and $\sum_{n=1}^\infty b_n$ with non-negative terms, we can use the following comparison: if $a_n\leq b_n \forall n$, then $\sum_{n=1}^\infty a_n\leq \sum_{n=1}^\infty b_n$ (Theorem 2.10. in Pete L. Clark's Sequences and Series: A Sourcebook).
Is there any way I could approach the problem of comparing two series (knowing only that they are both convergent, but not the limit of the partial sums), if their terms take alternating signs (or, more generally, some of them might be negative)?
Best Answer
I would say the same answer is "no". You basically need to distinguish two cases: if the series is absolutely convergent, you focus on one of the positive and negative parts separately, and you might use comparison for each.
When your series is conditionally convergent you have two main avenues to exploit: if the series is alternating, you would usually go for Leibnitz's Criterion. Otherwise the best choice is likely to be to group terms in twos or threes or more to get either as series of positive terms or an alternating one.