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Source. C. Badea, The irrationality of certain infinite series, Glasgow Math. J. 29 (1987), no. 2, 221--228, doi:10.1017/S0017089500006868. The Theorem is stated on p. 222 (Section 2, "Main result") and proved on pp. 222--223. Bibliographic details are on the source card.

Statement

Theorem (p. 222). Let $(b_n)$ and $(a_n)$, $n\ge1$, be sequences of positive integers such that

$$a_{n+1}>\frac{b_{n+1}}{b_n}\,a_n^2-\frac{b_{n+1}}{b_n}\,a_n+1$$

(the paper's (1)) holds for every large $n$. Then $\sum_{n\ge1}b_n/a_n$ is irrational.

The paper assumes throughout that the series it treats converge, or alternatively adopts the convention that $\infty$ counts as irrational (Section 1, p. 221); the Theorem is read under that convention.

The remark after the proof (pp. 223--224) notes that the same argument would give the Theorem for positive real $a_n$ and $b_n$ if Froda's generalization of Brun's criterion held, and states that Froda's generalization is false, citing the author's counterexample. The Theorem is stated and proved only for positive integers.

Proof pointer

Pp. 222--223. With $P_n=a_1\cdots a_n$ and $A_n=\sum_{j\le n}b_jP_n/a_j$, the partial sums are $A_n/P_n$, an increasing sequence of rationals. Brun's criterion (the paper's reference [3]) gives irrationality of the limit of an increasing sequence $y_n/x_n$ of quotients of positive integers when the difference quotients $(y_{n+1}-y_n)/(x_{n+1}-x_n)$ decrease strictly for all large $n$. Using $A_{n+1}=a_{n+1}A_n+b_{n+1}P_n$, the paper reduces that condition for $y_n=A_n$, $x_n=P_n$ to inequality (1).

Read depth. Claims checked: the statement and its hypotheses were read clause by clause on p. 222 of the print; the proof was read for structure only.

Dependencies

Brun's irrationality criterion (V. Brun, 1910, the paper's reference [3]), used as a black box.

Bears on

• Problem 267: through Corollary 1 the paper derives Corollary 4, the irrationality of $\sum_{n\ge1}1/F_{2^n+1}$, which is the single instance $n_k=2^k+1$ of the problem. The Theorem states nothing about other index sequences.
• Problem 243: through Corollary 1, a sequence of positive integers with rational reciprocal sum has $a_{n+1}\le a_n^2-a_n+1$ for infinitely many $n$; the paper does not give the problem's conclusion.
• Problem 263: context only. The problem page records a thread remark calling the Theorem a stronger classical irrationality criterion; the paper addresses neither of the problem's questions.