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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?

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You should do something about the first term. As it is, your series starts with $\frac{1}{0}$. – Dan Shved Mar 30 '14 at 6:04
    
@Dan Shved: Yep, thanks! – barak manos Mar 30 '14 at 6:04
    
Related: Sum of inverse of Fibonacci numbers – Martin Sleziak Mar 24 at 11:29
    
@MartinSleziak: That question was posted 6 hours ago. You should link my question there, not the other way round. – barak manos Mar 24 at 14:10
    
@MartinSleziak: Oh, I see you already did that... :) – barak manos Mar 24 at 14:11
up vote 8 down vote accepted

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).

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Thanks......... – barak manos Mar 30 '14 at 6:11

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