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Statement

Theorem 2 (p. 131). Let $\{n_k\}$ satisfy

• (i) $\{n_k^2/n_{k+1}\}$ is bounded;
• (ii) $\{N_k^*/n_{k+1}\}$ is bounded, where $N_k^*=n_1n_2\cdots n_k$.

If $\sum1/n_k$ is rational, then $\{n_k^2/n_{k+1}\}$ has only finitely many limiting values, all of them rational, and $\liminf n_k^2/n_{k+1}\le1$.

The statement does not repeat that $\{n_k\}$ is an increasing sequence of positive integers, as in Theorem 1; the context supplies it. Condition (ii) here, with the product in place of the lcm, is stronger than condition (ii) of Theorem 1, while (i) here is weaker than (i) there. The proof on p. 132 refers to this condition as (ii$'$).

Proof pointer

Pp. 131--132. Run the proof of Theorem 1 with $N_k^*$ in place of $N_k$: the bound on $d_k$ survives, and (6) becomes an equality $c_{k+1}=c_kn_{k+1}^2/n_{k+2}+o(1)$ (6$'$), so every limiting value of $n_k^2/n_{k+1}$ is a ratio of two of the bounded positive integers $c_k$, with numerator and denominator at most the bound of $bN_k^*/n_{k+1}$. If $\liminf n_k^2/n_{k+1}=1+\delta>1$, the telescoping product $N_k^*/n_{k+1}=\frac1{n_1}\cdot\frac{n_1^2}{n_2}\cdots\frac{n_k^2}{n_{k+1}}$ exceeds $C(1+\delta)^k$, contradicting (ii).

Dependencies

The proof of Theorem 1.

Source. P. Erdős and E. G. Straus, On the irrationality of certain Ahmes series, J. Indian Math. Soc. (N.S.) 27 (1964), 129--133; the edition read is named on the source card.

Read depth. Claims checked: the statement was read clause by clause on the page image of p. 131 and the proof on pp. 131--132 for its structure. Nothing here is independently reviewed.

Bears on

• Problem 243: background only. Theorem 2 concerns sequences where $n_k^2/n_{k+1}$ need not tend to $1$ and gives no recurrence, so it does not give the problem's conclusion for any class of sequences.