Well I was having a doubt on the infinite sum of the reciprocals of the Fibonacci series, that is,


Assuming that the series starts with $1$ rather than $0$. The series is obviously convergent as $$\lim_{n\to\infty}\{\frac{\frac{1}{F_{n+1}}}{\frac{1}{F_{n}}}\}<1$$ Therefore it must converge to some specific value. Can someone please help me find the value.

  • $\begingroup$ I'm fairly positive this can be found on the Wikipedia page.... A little research goes a long way $\endgroup$ – Brevan Ellefsen Jun 9 '16 at 15:33
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    $\begingroup$ en.wikipedia.org/wiki/Reciprocal_Fibonacci_constant $\endgroup$ – Jack D'Aurizio Jun 9 '16 at 15:34
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    $\begingroup$ I suppose the "re_s_iprocals" in the title is a typo? $\endgroup$ – 0x539 Jun 9 '16 at 15:34
  • $\begingroup$ To summarize the Wikipedia page I alluded to and Jack D'Aurizio linked to, , the answer is 3.359885... and the number can be shown to be convergent through the ratio test $\endgroup$ – Brevan Ellefsen Jun 9 '16 at 15:36
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    $\begingroup$ Further note that the number was proved irrational by Richard André-Jeannin $\endgroup$ – Brevan Ellefsen Jun 9 '16 at 15:37

Mathematica tells us that it is

(Sqrt[5]/(8 ArcCsch[2]) * 
(ArcCosh[3/2] EllipticTheta[2, 0, 1/GoldenRatio^2]^2 + 
 Log[5] + 2 QPolyGamma[0, 1, 1/GoldenRatio^4] - 
 4 QPolyGamma[0, 1, 1/GoldenRatio^2]

which it outputs in TraditionalForm as $$\frac{\sqrt{5} \left(2 \psi _{\frac{1}{\phi ^4}}^{(0)}(1)-4 \psi _{\frac{1}{\phi ^2}}^{(0)}(1)+\cosh ^{-1}\left(\frac{3}{2}\right) \vartheta _2\left(0,\frac{1}{\phi ^2}\right){}^2+\log (5)\right)}{8 \ \text{csch}^{-1}(2)}$$


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