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Find the limit on fibonacci numbers

Source: 2019 Jozsef Wildt International Math Competition-W. 17

May 18, 2020
Fibonacci sequencenumber theory

Problem Statement

Let fn=(1+1n)n((2n−1)!Fn)1nf_n=\left(1+\frac{1}{n}\right)^n\left((2n-1)!F_n\right)^{\frac{1}{n}}. Find lim⁡n→∞(fn+1−fn)\lim \limits_{n \to \infty}(f_{n+1} - f_n) where FnF_n denotes the nnth Fibonacci number (given by F0=0F_0 = 0, F1=1F_1 = 1, and by Fn+1=Fn+Fn−1F_{n+1} = F_n + F_{n-1} for all n≥1n \geq 1
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