# A Fibonacci series

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.

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

• There may be a more elegant way, but Binet's formula (google it if necessary) makes it easy. – Daniel Fischer Mar 28 '14 at 23:50

Let the sum be $S$.
Hint: What is the value of $S - \frac{1}{2} S - \frac{1}{4}S$
$S - \frac{1}{2} S - \frac{1}{4}S = \frac{1}{2}$