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Limit involving the fibonacci sequence

Source: Moldovan MO 2003

February 21, 2006
limitfunctioncalculuscalculus computations

Problem Statement

Let (Fn)n∈N∗(F_n)_{n\in{N^*}} be the Fibonacci sequence defined by F1=1F_1=1, F2=1F_2=1, Fn+1=Fn+Fn−1F_{n+1}=F_n+F_{n-1} for every n≥2n\geq{2}. Find the limit: lim⁡n→∞(∑i=1nFi2i) \lim_{n \to \infty}(\sum_{i=1}^n{\frac{F_i}{2^i}})
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