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Source. Corollary 2.10 and its proof, printed p. 87, with the closing remark of the section. Read on the page image.

Statement

Suppose $\{a_n\}$ and $\{b_n\}$ meet the hypotheses of Theorem 2.1 ($a_n>1$ for large $n$, $|b_n|/(a_{n-1}a_n)\to0$), and that from some point on $b_n$ is positive, $a_n$ is nondecreasing, and

$$\lim\frac{b_{n+1}-b_n}{a_n}\le0\qquad\text{and}\qquad\liminf\frac{a_n}{b_n}=0 .$$

Then the series $\sum b_n/(a_1\cdots a_n)$ is irrational. (The paper prints "$\lim$" in the first condition; the proof uses it as an upper limit, and Hančl–Tijdeman 2004 quote it as $\limsup\le0$.)

Closing remark (p. 87): without the hypothesis $\liminf a_n/b_n=0$, a rational sum is possible only if positive integers $B,C$ give $Bb_n=C(a_n-1)$ for every large $n$.

Proof structure (p. 87)

Rationality gives, by Theorem 2.1, $B$ and $c_n$ with $Bb_n=c_na_n-c_{n+1}$ and $c_{n+1}/a_n\to0$. Then $b_{n+1}/b_n>(c_{n+1}-\varepsilon)/c_n$ for large $n$, so $c_{n+1}>c_n$ would give (2.11) $b_{n+1}>b_n+(1-\varepsilon)^2a_n/B$, contradicting $\lim(b_{n+1}-b_n)/a_n\le0$. So from some point on $0<c_{n+1}\le c_n$; the $c_n$ are then bounded, and so is $b_n/a_n$ (as $Bb_n<c_na_n$), which $\liminf a_n/b_n=0$ forbids.

Specialization to the prime factorial series

Take $a_n=n$ and $b_n=p_n$. Then $a_n>1$ for $n\ge2$; $p_n/((n-1)n)\to0$ and $\liminf n/p_n=0$ follow from $p_n\sim n\log n$; $b_n>0$ and $a_{n+1}\ge a_n$ hold; and $\lim(p_{n+1}-p_n)/n\le0$ is the gap bound $p_{n+1}-p_n=o(n)$, which the prime number theorem with remainder supplies (as on the density page of the 1958 card). So $\sum p_n/n!$ is irrational: a reproof of the case $k=1$ of Erdős 1958, with the same external input. For $b_n=p_n^k$, $k\ge2$, the hypothesis $b_n/(a_{n-1}a_n)\to0$ of Theorem 2.1 fails, so the corollary says nothing about the higher powers.

Bears on. #251 (context: reproof of the $k=1$ theorem cited on the problem page; nothing on $\sum p_n/2^n$, where $a_n=2$ violates the hypothesis (2.2) of Theorem 2.1).