What is the sum of Fibonacci reciprocals?

Question

How can I calculate $\sum\limits_{n=1}^{\infty}\frac{1}{F_n}$, where $F_0=0$, $F_1=1$ and $F_n=F_{n-1}+F_{n-2}$?

Empirically, the result is around $3.35988566$.

Is there a "more mathematical way" to express this?

Answer 1

This is A079586, where you can find several references. It doesn't look like there is a 'nice' closed form, but some results have been proved. The constant is irrational [1] and can be computed rapidly [2], [3] with various methods.

[1] Richard André-Jeannin, Irrationalité de la somme des inverses de certaines suites récurrentes, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics 308:19 (1989), pp. 539-541.

[2] Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).

[3] William Gosper, Acceleration of Series, Artificial Intelligence Memo #304 (1974).

Answer 2

It has a closed form formula given by Vladimir Reshetnikov in year $2015$ $$\sum_{i=1}^\infty \frac{1}{F_n}=\frac{\sqrt{5}}{4}\Bigg[\frac{2 \psi _{\frac{1}{\phi ^4}}(1)-4 \psi _{\frac{1}{\phi ^2}}(1)+\log (5)}{2 \log (\phi )}+\vartheta _2\left(0,\frac{1}{\phi ^2}\right){}^2 \Bigg]$$ where appear the q-digamma function and the theta function.