# Sum of Reciprocals of the Fibonacci Series

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

$$S=1+1+\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+....$$

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.

• I'm fairly positive this can be found on the Wikipedia page.... A little research goes a long way – Brevan Ellefsen Jun 9 '16 at 15:33
• en.wikipedia.org/wiki/Reciprocal_Fibonacci_constant – Jack D'Aurizio Jun 9 '16 at 15:34
• I suppose the "re_s_iprocals" in the title is a typo? – 0x539 Jun 9 '16 at 15:34
• 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 – Brevan Ellefsen Jun 9 '16 at 15:36
• Further note that the number was proved irrational by Richard André-Jeannin – Brevan Ellefsen Jun 9 '16 at 15:37

(Sqrt[5]/(8 ArcCsch[2]) *

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