Simple question, if we have a sum that has a known closed form such as the following. $$\sum_{k=0}^{n}{a_k}=S_n$$ Then can we express, $$\sum_{k=0}^{n}{(-1)^ka_k}$$ in terms of $S_n$? If so, when? I have tried the obvious, such as trying to construct the original sum using the alternating one, but to no end since all terms will cancel out.
I did not find anything about this on here so if anybody has any input on this or any references I'd be thankful.