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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. Corollary 5 and its proof are on p. 227 (Section 5). Bibliographic details are on the source card.

Statement

The Lucas numbers are $L_n=F_{n-1}+F_{n+1}$, with $F_n$ the Fibonacci numbers, $F_0=0$, $F_1=1$ (pp. 226--227).

Corollary 5 (p. 227, quoted). "The sum of the series $\sum_{n=1}^{\infty}1/L_{2^n}$ is an irrational number."

Section 5 presents this as the answer to the second of the two questions Erdős and Graham raise in Old and new problems and results in combinatorial number theory (1980), pp. 64--65.

Proof pointer

P. 227. The paper shows $L_{2p}>L_p^2-L_p+1$ for all large $p$ (its (12)), reducing it through $F_{2k+1}=F_k^2+F_{k+1}^2$ to the inequality (13), which holds for large $p$ by $F_p^2-F_{p+1}F_{p-1}=(-1)^{p+1}$. Taking $p=2^n$ gives (8) for $a_n=L_{2^n}$, and Corollary 1 applies.

Read depth. Claims checked: the statement was read on p. 227 of the print and the short proof was followed.

Dependencies

Corollary 1 and the two Fibonacci identities above.

Bears on

• Problem 267: context only. The series is over Lucas numbers, not Fibonacci numbers, so it is not an instance of the problem.