# Thread: Relation between a series and its alternating version.

1. ## Relation between a series and its alternating version.

Hello! Here is something that I've been wondering for quite a while. Consider the following two infinite series:

$S = \sum_{k\ge0}\frac{1}{(ak+b)(ck+d)}, ~~ T = \sum_{k\ge0}\frac{(-1)^k}{(ak+b)(ck+d)}$

Assuming that all issues of convergence are sorted, what's the relation (if any) between the values of S and T?

2. I don't know if I have correctly understood your question. In general, if $u_k\geq 0$ for all k and $\sum_{k\geq 0}u_k,\;\sum_{k\geq 0}(-1)^ku_k$ are both convergent (with sums S and T respectively) then, it is clear that $\sum_{k\geq 0}u_{2k}$ is convergent (denote by R its sum). Then, according to the algebra of series : S+T=R .

3. Originally Posted by FernandoRevilla
I don't know if I have correctly understood your question. In general, if $u_k\geq 0$ for all k and $\sum_{k\geq 0}u_k,\;\sum_{k\geq 0}(-1)^ku_k$ are both convergent (with sums S and T respectively) then, it is clear that $\sum_{k\geq 0}u_{2k}$ is convergent (denote by R its sum). Then, according to the algebra of series : S+T=R .
I think you have understood my question correctly, thank you. What about $\sum_{k\ge 0}(-1)^ku_{2k}$? I think it's convergent, but I'm not sure how it relates S and T. Basically, what I wanted to see if there is a way to find S and T by the just finding one of them.

4. Originally Posted by TheCoffeeMachine
Basically, what I wanted to see if there is a way to find S and T by the just finding one of them.

Consider the series:

$(i)\quad 0+\dfrac{1}{1^2}+0+\dfrac{1}{2^2}+0+\dfrac{1}{3^2} +\ldots$

$(ii)\quad \dfrac{1}{1^2}+0+\dfrac{1}{2^2}+0+\dfrac{1}{3^2}+0 +\ldots$

In both cases $S=\pi^2/6$ . However $T=-\pi^2/6$ in the first case and $T=\pi^2/6$ in the second one.