# Sum of Power of Two Fibonacci reciprocals [duplicate]

Evaluate $$\sum_{k=0}^{\infty} \frac{1}{F_{2^n}} \;.$$

I'm thinking of using a relation from a term to another.

## marked as duplicate by Martin Sleziak, Workaholic, Watson, Ethan Bolker, Davide GiraudoNov 6 '16 at 19:56

This series is known as the Millin series. One can show that, $$\sum_{n=0}^\infty\frac{1}{F_{2^n}}=\frac{7-\sqrt{5}}{2}$$ by first proving, using induction, that: $$\sum_{n=0}^\ell \frac{1}{F_{2^n}}=3-\frac{F_{2^\ell-1}}{F_{2^\ell}},$$ and then taking the limit as $\ell\to\infty$, knowing that $\lim\limits_{\ell\to\infty}\dfrac{F_{\ell+1}}{F_\ell}=\frac{1+\sqrt{5}}{2}$, $$\sum_{n=0}^\infty\frac{1}{F_{2^n}}=\lim_{\ell\to\infty}\left(3-\frac{F_{2^\ell-1}}{F_{2^\ell}}\right)=3-\dfrac{1}{(1+\sqrt{5})/2}=\dfrac{7-\sqrt{5}}{2}.$$