In a Fibonacci sequence 1, 1, 2, 3, 5, …, Fn, Fn+1
define Rn = Fn/ Fn-1
Prove that lim (Rn
) as n approaches infinity
a.k.a. the golden ratio, φ (phi).
(In reply to re: Solution
by Steve Herman)
Consider, for instance, the logistic equation, f(n+1) = rf(n)*(1-f(n)), where f(n) is between 0 and 1.
Does it converge as n goes to infinity?
It is easy to calculate that if it has a limit, then the limit must be 1 - 1/r.
However, surprisingly, it does not have a limit if r is large enough, a finding which helped jumpstart the mathematical discipline of chaos theory.
For instance, if r = 3.5 and f(0) = .4, it starts repeating the numbers .3828, .8269, .5009, .6750, .3828, etc., and never converges
In other words, being able to calculate the only possible limit does not prove that the series converges. I will try this weekend to prove that the Fibonacci ratio actually does in fact converge to its only possible limit.