A Fibonacci series

Question

Let $F_n$ be the $n^{th}$ term of the Fibonacci sequence. That is, $F_1 = F_2 = 1$ and $F_n$ is defined recursively for $n\geq3$ by $F_n = F_{n-2}+F_{n-1}$. It is a known fact that $$ \lim_{n\to\infty}\frac{F_{n+1}}{F_n}=\frac{1+\sqrt{5}}{2} $$

show that the series $$ \sum_{n=1}^{\infty}\frac{F_n}{2^n} $$

is convergent and compute its sum.

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By the ratio test, I know the series is convergent. However, I have difficulty evaluating it. Any hints would be very helpful! Thanks.

Answer 1

Let the sum be $S$.

Hint: What is the value of $S - \frac{1}{2} S - \frac{1}{4}S $

Think about the usual way you deal with a geometric progression.

$S - \frac{1}{2} S - \frac{1}{4}S = \frac{1}{2} $