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Source. C. Badea, The irrationality of certain infinite series, Glasgow Math. J. 29 (1987), no. 2, 221--228, doi:10.1017/S0017089500006868. The definitions and the Proposition are on p. 225 (Section 4); the proof and a remark are on p. 226. Bibliographic details are on the source card.

Statement

Definitions (p. 225). A subseries of $\sum u_n$ is a series $\sum v_n$ with $(v_n)$ a subsequence of $(u_n)$. Two subseries $\sum v_n$ and $\sum w_n$ of the same series are disjoint when $(v_n)$ and $(w_n)$ are disjoint subsequences of $(u_n)$.

Proposition (p. 225, quoted). "Every convergent infinite series of positive rationals has infinitely many disjoint subseries with irrational sums."

The remark on p. 226 notes, citing Călin and Kiss (1970), that some convergent series of positive rationals has every subseries with an irrational sum, so "irrational" cannot be replaced by "rational" in the Proposition.

Proof pointer

P. 226. Write the series as $\sum b_n/a_n$ with positive integers $a_n$, $b_n$; the terms tend to $0$. The paper chooses infinitely many disjoint index subsequences along which the condition of the paper's Corollary 3 holds for all large $k$, and applies Corollary 3 (p. 225), a consequence of the Theorem, to each.

Read depth. Claims checked: the definitions and the statement were read clause by clause on p. 225 of the print; the proof was read for structure only.

Dependencies

Corollary 3 (p. 225) and through it the Theorem.

Bears on

None: the paper relates the Proposition to no Erdős problem, and no problem page cites it.