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Quite a bit of research has been done on the question of whether certain infinite series - consisting of rational summands - amount to irrational numbers. Examples include this article by Erdős (1975, PDF), this one by Badea (1987), this paper by Tachiya (2004), and finally an article by Borwein (2008).

I am curious about a different case, where one considers infinite series of irrational summands. Can one establish any criteria or conditions for the (ir)rationality of such sums?

The following MSE question considers infinite sums of irrational numbers that amount to a rational number. Examples include

$$\tan \left( \frac{\pi}4 \right)=\sum_{n=0}^\infty \frac{(-1)^n 2^{2n+2}(2^{2n+2}-1)B_{2n+2}}{(2n+2)!}\left(\frac{\pi}4\right)^{2n+1}=1 \tag{1}\label{1}$$

and, as I describe in my answer through a rational zeta series:

$$ \sum_{n=1}^{\infty} \left(\zeta(2n)-1 \right) = \frac{3}{4}. \tag{2}\label{2}$$

Please note that we consider infinite series here, not finite sums of irrational numbers.

My question is:

Are there any results and resources in the literature on establishing conditions or criteria for the (ir)rationality of infinite series consisting of irrational summands?

Max Lonysa Muller
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  • Since irrationality manifests itself in so many different ways -- none of "NOT being a quotient of integers" or "NOT having a terminating decimal expansion" or "NOT having a finite continued fraction expansion" provides much structure/properties that can be worked with -- I don't see that much could be said unless the irrational numbers are restricted so as to all satisfy some specific property (e.g. each possible digit string appears in the decimal expansion, are all normal numbers, are all quadratic irrationals, are all algebraic numbers, etc.). – Dave L. Renfro Oct 30 '24 at 16:50
  • @DaveL.Renfro Do you think the question becomes more feasible if one requires all irrational numbers (the summands) to have the property that they are transcendental? – Max Lonysa Muller Oct 30 '24 at 16:53
  • Requiring them to all be transcendental seems to have the same difficulty, since we're essentially saying anything goes unless they're very specifically restricted to satisfy the extremely strong requirement of being an algebraic irrational. In pretty much every sense of "smallness" for sets of real numbers, all but an infinitesimally small set of real numbers are transcendental. – Dave L. Renfro Oct 30 '24 at 16:56

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