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Source. Theorem 4.3, preprint p. 9, with its proof (p. 9), the sentence before it (p. 8) and the Remark after it (p. 9). 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 with

$$a_nb_{n+1}-a_{n+1}b_n\le b_{n+1}-b_n\qquad\text{for all large }n.$$

Then $\sum_{n\ge1}b_n/(a_1\cdots a_n)$ is rational if and only if $(a_n-1)/b_n$ is constant for $n\ge n_0$.

The hypothesis can be rewritten as $b_{n+1}(a_n-1)\le b_n(a_{n+1}-1)$; when every $a_n>1$ this is $b_{n+1}/(a_{n+1}-1)\le b_n/(a_n-1)$, the form in which the abstract (p. 1) states criterion (ii). The sentence before the theorem (p. 8) presents it as a variant of Theorem 4.2 in which $a_n$ need not be monotonic, with a proof of a different structure. As throughout the paper, convergence of the series is assumed when its rationality is discussed (p. 2).

Proof pointer (p. 9)

One direction follows from Lemma 2.1. For the other, with $S=r/q$: if the remainders $R_n$ are eventually nonincreasing, the end of the proof of Theorem 4.2 applies; an index with $R_{m+1}>R_m$ is ruled out by an identity (the paper's (9)) derived from $R_{n+1}=a_nR_n-b_n$, together with the hypothesis, which make the increments grow and force $b_n/a_n\to0$, and then $qR_n\in\mathbb{Z}$ forces $R_n=0$, which is impossible.

Consequences in the paper

The Remark after the proof (p. 9) derives Badea's criterion, case (i) of Corollary 4.1, from this theorem by rewriting the Ahmes-type series as a Cantor series, and Corollary 4.2 (pp. 9--10) is obtained in a similar way.

Bears on. No catalog problem directly; its consequences Corollary 4.1 and Corollary 4.2 carry the relation to problem 243.