Looking for some references or hints on the following:

Suppose we have a strictly increasing sequence of positive integers $\alpha=\{a_k\}_{k=1}^{n}$ and we define the sum $$ S_\alpha=\sum_{k=1}^n\frac{1}{a_k^2}. $$ Clearly when $\alpha$ has a finite number of terms, i.e. when $n<\infty$, then $S_\alpha$ is rational. And when $\alpha=\mathbb{N}$ we have $S_\alpha=\frac{\pi^2}{6}$, which is irrational.

I am interested in finding infinite sequences for which $S_\alpha$ is rational, if there are any. Does anyone know of a good source where this idea might be discussed?

  • 3
    $\begingroup$ $a_k=2^k$....... $\endgroup$ – Rene Schipperus Apr 17 '18 at 2:03
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    $\begingroup$ Hurwitz's Theorem suggests there cannot be very many such infinite sequences. $\endgroup$ – Eric Towers Apr 17 '18 at 2:04
  • $\begingroup$ Hurwitz's theorem has hardly anything to do with this. $\endgroup$ – Ivan Neretin Apr 17 '18 at 8:19
  • $\begingroup$ @ReneSchipperus Thank you. I thought of the geometric series shortly after I posted this, but I wonder if there any arithmetic sequences that would produce a rational sum. $\endgroup$ – Laars Helenius Apr 17 '18 at 14:16
  • $\begingroup$ @EricTowers I agree with Ivan. I don’t see how Hurwitz’s Theorem says anything about this. If you have some time, could you elaborate? $\endgroup$ – Laars Helenius Apr 17 '18 at 14:17

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