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Assume we have a sequence of positive rational numbers $(a_n)$, and $\sum_{n=1}^\infty a_n = x$ and $x$ is transcendental. If we have a subsequence of $(a_n)$, $(b_n)$ and $\sum_{n=1}^\infty b_n = y$ and $y$ is not a rational number, is $y$ also transcendental? Further, if $y$ is transcendental, is it of the form $x-s,$ where $s$ is a rational number?

Take as an example the Basel problem. We have that $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}.$ This is a transcendental number. The sum of every finite subsequence of $(\frac{1}{n^2})$ is rational and there are infinitely many infinite subsequences that sum to a rational number, for example, $(\frac{1}{(p^n)^{2}})$ where $p$ is a prime.

What if we have a subsequence whose sum is not a rational number? Say $(k_n)$ is such a sequence. Is the sum $\sum_{n=1}^\infty k_n$ transcendental? Is $\sum_{n=1}^\infty k_n = \frac{\pi^2}{6}-s$, where $s\in\mathbb Q$? Or could we select the members very carefully and get, say, $\sqrt{2}?$

I figured that if we see the reciprocals of squares as a set $A$, and let $B = \{z \in P(A)\ |\ \sum_{i \in z} i \in \mathbb Q\}$, then $B$ is an ideal, so the set of subsets whose sum is irrational ($P(a)\setminus B$) should be filter. But I doubt it is much use.

Valtteri
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    No. Take two rational sequences whose series converge absolutely to $\sqrt 2$ and $\pi - \sqrt 2$, respectively. The sequence that alternates between these two sequences will sum to $\pi$ which is transcendental but we have an explicit subsequence converging to $\sqrt 2$ which is irrational without being transcendental. – User8128 Sep 05 '18 at 22:05
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    For your second question, $\sum_{n=1}^\infty \frac{1}{(2n)^2} = \frac{\pi^2}{24}$, yet $\pi^2/6-\pi^2/24 = \pi^2/8$ is irrational – Jakobian Sep 05 '18 at 22:06

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In this answer, I prove that there is a subsequence of $\sum \frac{1}{n^2}$ which sums to $x$, for any $x \in \left(0, \frac{\pi^2}{6} - 1\right] \cup \left(1, \frac{\pi^2}{6}\right]$. You can produce such a subsequence greedily: that is, at each step, always take the largest remaining term which doesn't cause you to exceed your target.

For example, this method gives us a series $$ \sqrt{2}=1+1/2^2+1/3^2+1/5^2+1/9^2 + 1/37^2+1/195^2+1/8584^2+1/1281816^2+\dots $$ assuming I haven't run into any roundoff errors.

Micah
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  • In my question https://math.stackexchange.com/questions/311695, I was given examples that demonstrate that there are sequences of rational numbers without any subsequences that sum to a rational number. Does the theorem in your answer hold for them? – Valtteri Sep 05 '18 at 22:20
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    @Valtteri: No. Basically, the theorem in my answer is about sums whose terms go to zero sufficiently slowly (roughly, more slowly than $2^{-n}$). The sequences in your other answer go to zero much faster than that! – Micah Sep 05 '18 at 22:26
  • Ah, I see. Very curious indeed. Clearly my intuition isn't very good when it comes to this. – Valtteri Sep 05 '18 at 22:30
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    One more thing. I once asked this question about when a subset of reciprocals of natural numbers is able to produce every number between zero and one without repeating a term https://math.stackexchange.com/questions/1247191, which has no satisfactory answer. Would you say that you answer could be used to answer it? – Valtteri Sep 05 '18 at 22:42
  • It definitely gives you one direction: any sequence $a_n$ with $\sum \frac{1}{a_n} \geq 1$ and $\frac{1}{a_n} < \sum_{k=n+1}^\infty \frac{1}{a_k}$ for all $n$ has the property you want. I'd have to think some more about the other direction (i.e., whether every function with your property also satisfies my inequalities). I may post an answer over the next few days depending on whether I come up with anything... – Micah Sep 06 '18 at 02:55