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Claim. D. Duverney, Irrationality of fast converging series of rational numbers, J. Math. Sci. Univ. Tokyo 8 (2001), 275--316, received 13 September 2000. Corollary 3.2 (printed p. 287) states: let $u_n$ be positive integers with

$$\lim_{n\to\infty}u_n=+\infty \quad\text{and}\quad \sum_{n=0}^{\infty}\Bigl(\frac{u_{n+1}}{u_n^2}-1\Bigr)\ \text{convergent},$$

and let $a_n\in\{-1,1\}$ for every $n$; then $\sum_{n\ge0}a_n/u_n$ is rational if and only if

$$u_{n+1}=u_n^2-\frac{a_{n+1}}{a_n}u_n+\frac{a_{n+2}}{a_{n+1}}$$

for every $n\ge N$. With every $a_n=1$ this is $u_{n+1}=u_n^2-u_n+1$. The paper introduces the corollary as a partial answer to its question (2.15), Erdős's question of Problem 243, which it locates at p. 64 of the Erdős--Graham monograph and p. 105 of Erdős's 1988 survey, and proves it (Section 5.2) by Mahler's method in the form of Loxton and van der Poorten, the paper's tool for fast converging series. The source card duverney_2001_irrationality_fast_converging_series_rational_numbers records the publication.

Covers. The sequences of Problem 243 for which $\sum_n(a_{n+1}/a_n^2-1)$ converges; the convergence implies the problem's hypothesis $a_{n+1}/a_n^2\to1$, so these are instances of the problem, and the corollary decides them, with the recurrence as the exact condition for rationality. Not covered: the sequences whose relative error tends to $0$ without being summable.

Acceptance. Refereed: J. Math. Sci. Univ. Tokyo 8 (2001), 275--316. The site labels the problem OPEN and records the corollary in its remark, added after a thread comment of 11 October 2025 pointed to it, so no reviewed evidence is listed. The proof is not checked here.

Depends on. Nothing in this wiki.