Sum over subsequences of integers

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?

• $a_k=2^k$....... – Rene Schipperus Apr 17 '18 at 2:03
• Hurwitz's Theorem suggests there cannot be very many such infinite sequences. – Eric Towers Apr 17 '18 at 2:04
• Hurwitz's theorem has hardly anything to do with this. – Ivan Neretin Apr 17 '18 at 8:19
• @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. – Laars Helenius Apr 17 '18 at 14:16
• @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? – Laars Helenius Apr 17 '18 at 14:17