Given a convergent alternating series, what conditions can be placed on the terms in the summand?

Question

I have a sum $$S = \sum_{n=0}^\infty(-1)^na_n$$ which I know to be convergent--in fact, $S\leq 1$. I additionally know that $\forall n,\ a_n > 0$. Given finiteness of $S$, what conditions can be placed on the terms $\{a_n\}$?

This question is answered the other direction in a multitude of posts, which generally appeal to either instances or counterexamples to the alternating series test. However, it is not clear to me what, if any, conditions can be placed on the terms given convergence of the series.

For completeness, I will mention that in the context of this problem we can consider a deformation: $S(s) = \sum (-1)^n a_n s^n $ should also be finite for $\forall s > 0 $. I selected the $s=1$ case for convenience, but perhaps properties (unbeknownst to me) of this deformation allow for more constraints to be placed on the $a_n$.

Answer 1

A few considerations... Not sure this is what you're looking for though!

• For sure the sequence $\{a_n\}$ is converging to $0$.
• In case of alternating series, $\{a_n\}$ is decreasing.
• However, $\{a_n\}$ may not even be eventually decreasing.
• $\sum (-1)^na_n$ may be absolutely convergent or not.
• If not, reordering the terms, you can let the sum be equal to whatever you want.