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Source. Theorem 4.1, preprint p. 6; proof pp. 6--7; the paragraph opening Section 4 (p. 6). Read on the rendered pages. The paper is cited by its record on the source card.

Statement

Let $(a_n)_{n\ge1}$ and $(b_n)_{n\ge1}$ be sequences of positive integers for which $S:=\sum_{n\ge1}b_n/a_n$ converges, and let $A_n=\operatorname{lcm}(a_1,\ldots,a_n)$. Suppose

$$\limsup_{n\to\infty}A_{n-1}\Bigl(\frac{b_{n+1}a_n}{a_{n+1}}-\frac{b_n}{a_n}\Bigr)\le0.$$

Then $S$ is rational if and only if

$$a_{n+1}=\frac{b_{n+1}}{b_n}a_n(a_n-1)+1\qquad\text{for large }n.$$

The paper says (p. 6) that the proof is based on the proofs of Erdős and Straus (J. Indian Math. Soc. 27 (1964), 129--133) but is much simpler and more general.

Proof pointer (pp. 6--7)

For $S=r/q$ the proof works with the tails $R^\star_n=\sum_{k>n}b_k/a_k$, for which $qA_nR^\star_n$ is a positive integer. The hypothesis makes $a_nR^\star_n-R^\star_{n-1}$ smaller than $2\epsilon/A_{n-1}$ for large $n$; with $\epsilon=1/(4q)$ an integrality argument makes the products $a_1\cdots a_nR^\star_n$ eventually nonincreasing, hence eventually constant, and the recurrence follows. The converse direction is a telescoping identity for the partial sums from the index where the recurrence starts.

Relation to problem 243

With $b_n=1$ the hypothesis reads $\limsup(A_{n-1}/a_n)(a_n^2/a_{n+1}-1)\le0$ and the conclusion is the recurrence $a_{n+1}=a_n^2-a_n+1$ for large $n$ that problem 243 asks for. The problem's hypothesis $a_{n+1}/a_n^2\to1$ makes the second factor tend to $0$ but places no bound on $A_{n-1}/a_n$, so it does not imply the theorem's hypothesis: the theorem settles the problem only for sequences that also satisfy that condition. The paper derives its Corollary 4.1 from this theorem.

Bears on. #243 (context: with $b_n=1$ the conclusion is the problem's recurrence, under a hypothesis the problem's does not imply).