Evaluate the sum: $\sum\limits_{n=0}^{\infty} \frac1{F_{(2^n)}}$ [duplicate]

Evaluate the sum:

$$\sum_{n=0}^{\infty} \frac{1}{F_{(2^n)}}$$

where $F_{m}$ is the $m$-th term of the Fibonacci sequence. I need some support here. Thanks.

As wikipedia claims the result follows from the identity $$ \sum\limits_{n=0}^N\frac{1}{F_{2^n}}=3-\frac{F_{2^N-1}}{F_{2^N}} $$ You can try to prove it by induction.

See http://oeis.org/A079585 and references there.

Typing Sum[1/Fibonacci[2^n],{n,0,infty}] in http://www.wolframalpha.com gives,

$$\sum_{n=0}^\infty \frac{1}{F_{(2^n)}} = \left(\frac{1-\sqrt{5}}{2}\right)^3+\left(\frac{1+\sqrt{5}}{2}\right)^2 = 2.381966\dots$$