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Math Help - Relation between a series and its alternating version.

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    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?
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    MHF Contributor FernandoRevilla's Avatar
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    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 .
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    Quote Originally Posted by FernandoRevilla View Post
    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.
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    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by TheCoffeeMachine View Post
    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.
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