TITLE: What is the meaning of limit of Fibonacci sequence?
QUESTION [9 upvotes]: For general Fibonacci sequence with $F_1=F_2=1$,
It is known that limit of $\frac {F_{n+1}} {F_n} $ exists.
I am wondering what this limit implies and why it is important to compare these two sequences.
Thanks

REPLY [4 votes]: The recurrence $F_{i+2}=F_i+F_{i+1}$ gives rise to a simple recurrence relation for the ratio of successive terms $\frac{F_{i+1}}{F_i}\in\Bbb R\cup\{\infty\}$, namely $\frac{F_{i+2}}{F_{i+1}}=\frac{F_i}{F_{i+1}}+\frac{F_{i+1}}{F_{i+1}}=(\frac{F_{i+1}}{F_i})^{-1}+1$, which is of order$~1$. The dynamics of the map $x\mapsto x^{-1}+1$ on $\Bbb R\cup\{\infty\}$ is quite simple: the map has two fixed points $\frac{1+\sqrt5}2$ and $\frac{1-\sqrt5}2$, of which the first is attractive, ad the second is repulsive. For any sequence satisfying the recursion and with initial ratio not exactly equal to the second fixed point, the ratio of successive terms will converge to the the first fixed point as $n\to\infty\,$; this is in particular the case for the Fibonacci sequence.