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Source. Theorem 2.1 and its proof, preprint p. 4, with the preceding paragraph on Oppenheim [6]. Read on the rendered page.

Statement

For integer sequences with every $a_n>1$ and $b_n=O(a_n)$, if some irrational $\alpha$ is a limit point of the ratios $b_n/a_n$, then the (convergent) series $S=\sum_{n\ge1}b_n/(a_1\cdots a_n)$ is irrational.

Corollary 2.1 (p. 4): if $\lim b_n/a_n$ exists and is irrational, $S$ is irrational. Oppenheim (Amer. Math. Monthly 61 (1954), 235--241) had the same conclusion under $0\le b_n<a_n$.

Proof (p. 4), summarized

If $S=r/q$, then $qR_n\in\mathbb{Z}$ for all $n$ (Lemma 2.1). Along a subsequence $b_{n_k}/a_{n_k}\to\alpha$; since $\alpha\notin\mathbb{Q}$, $a_{n_k}\to\infty$. From $R_{n_k}=b_{n_k}/a_{n_k}+R_{n_k+1}/a_{n_k}$ and $|R_{n_k+1}|\le M(1+1/2+1/4+\cdots)=2M$ (with $|b_n/a_n|\le M$), the integers $qR_{n_k}$ tend to $q\alpha$, so $\alpha$ is rational, a contradiction. $\blacksquare$

Role

Not used for the prime series of this library's problems: for $a_n=2$ and $b_n=p_n$ the hypothesis $b_n=O(a_n)$ fails, and for $a_n=n$, $b_n=p_n$ the ratios $p_n/n$ tend to infinity. Recorded for the card's coverage.

Bears on. No catalog problem directly.