To solve the problem of finding the sum ( \sum_{n=1}^{\infty} \frac{F_n}{2^n} ) where ( F_n ) is the Fibonacci sequence (( F_1=1, F2=1, F{n}=F{n-1}+F{n-2} )), we use the generating function of Fibonacci numbers.

Step 1: Define the generating function

Let ( S = \sum_{n=1}^{\infty} F_n x^n ). For ( x=\frac{1}{2} ), we need to find ( S ).

Step 2: Derive the generating function

Using the Fibonacci recurrence relation:
[ S = F_1x + F_2x^2 + F_3x^3 + \dots ]
[ xS = F_1x^2 + F_2x^3 + \dots ]
[ x^2S = F_1x^3 + \dots ]

Subtract ( xS + x^2S ) from ( S ):
[ S(1 - x - x^2) = F_1x \quad (\text{higher terms vanish due to recurrence}) ]

Since ( F_1=1 ):
[ S = \frac{x}{1 - x - x^2} ]

Step 3: Substitute ( x=\frac{1}{2} )

[ S = \frac{\frac{1}{2}}{1 - \frac{1}{2} - \left(\frac{1}{2}\right)^2} = \frac{\frac{1}{2}}{\frac{1}{4}} = 2 ]

Answer: (\boxed{2})

作者声明:本文包含人工智能生成内容。