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Source. I. J. Good, A reciprocal series of Fibonacci numbers, Fibonacci Quarterly 12 (1974), no. 4, p. 346. The paper's single, unnumbered Theorem and its proof are both on p. 346. Bibliographic details are on the source card.

Statement

With $F_1=F_2=1$ and $F_{n+1}=F_n+F_{n-1}$ the Fibonacci numbers, the paper states (p. 346)

$$\frac{1}{F_1}+\frac{1}{F_2}+\frac{1}{F_4}+\frac{1}{F_8}+\frac{1}{F_{16}}+\cdots=\frac{7-\sqrt5}{2}.$$

The indices are the powers $2^k$, $k\ge0$, each taken once. The paper does not remark that the value is irrational; it is a quadratic irrational because $\sqrt5$ is irrational.

Proof pointer (p. 346)

The proof rests on the finite identity

$$\frac{1}{F_1}+\frac{1}{F_2}+\frac{1}{F_4}+\cdots+\frac{1}{F_{2^n}}=3-\frac{F_{2^n-1}}{F_{2^n}},$$

which the paper says follows by induction with Binet's formula; it does not state the range of $n$, and the identity holds for every $n\ge1$ (for $n=1$ both sides equal $2$). Letting $n\to\infty$, the ratio $F_{2^n-1}/F_{2^n}$ tends to $(\sqrt5-1)/2$, which gives the theorem. The paper prints no further detail.

Read depth. Claims checked: the statement and the identity were read on p. 346 of the printed note; the identity was checked here for $n=1,2$ only.

Dependencies

Binet's formula for $F_n$.

Bears on

• Problem 267: the theorem is the instance $n_k=2^{k-1}$ ($n_1=1$, $n_2=2$, $n_3=4$, ...), whose ratio $n_{k+1}/n_k$ is exactly $2$, and it shows that this one sum is irrational by evaluating it as $(7-\sqrt5)/2$. The paper does not pose or mention the general question and says nothing about any other index sequence.