TITLE: Use of the Reciprocal Fibonacci constant?
QUESTION [8 upvotes]: The Reciprocal Fibonacci constant ($\psi$) is defined as
$$\psi=\sum_{k=1}^{\infty} \frac{1}{F_k}$$
where $F_{k}$ is the $k^{th}$ Fibonacci number.
The irrationality of $\psi$ has been proven.  Does the Reciprocal Fibonacci constant have a use in mathematics or is is notable simply because it is the value of an interesting sum?

REPLY [4 votes]: As far as I know, the constant $\psi$ doesn't arise naturally other than in just studying it from the outset. However, a related constant does make its appearance in a surprising way. That is, $$\beta:=1+\sum_{k=1}^\infty \dfrac{1}{\mathrm F_{2k}},$$or rather its reciprocal $1/\beta\approx0.3944196702$, arises as the solution to a problem in the analysis of bounded real sequences, which is the title question of my paper that addresses it: "How slowly can a bounded sequence cluster?". It is fully answered there and in the follow-up paper "A maximally separated sequence". The two papers are published in Functiones et Approximatio, volumes 46.2 (2012) and 48.1 (2013), which you can read if you have access to a library that subscribes to Project Euclid (the abstracts are freely available on-line). When I set out on the research, I had no inkling that a number with this sort of structure would come up. If this interests you, I can provide more details.