TITLE: Evaluate the sum: $\sum\limits_{n=0}^{\infty} \frac1{F_{(2^n)}}$
QUESTION [10 upvotes]: 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.

REPLY [6 votes]: 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.