Fibonacci limit

Calculus Level 1

The Fibonacci sequence can be defined with induction as Fn=Fn1+Fn2F_n=F_{n-1}+F_{n-2}, where F1=0F_1=0 and F2=1F_2=1. It is also well-known that the limit of the ratio between two consecutive terms limFn+1Fn\lim \frac{F_{n+1}}{F_n} is the golden ratio ϕ\phi.

In the generalized Fibonacci sequence below, the original Fibonacci sequence is Fn2.F_n^2. Fnm=i=1mFnim  with  Fnm=0  if  n<m  and  Fmm=1.F_n^m=\sum_{i=1}^{m}{F_{n-i}^m}\ \text{ with }\ F_n^m=0\ \text{ if } \ n<m\ \text{ and }\ F_m^m=1. Every sequence will also have a generalized golden ratio ϕm=limnFn+1mFnm.\phi_m=\displaystyle \lim_{n \rightarrow \infty}{ \frac{F_{n+1}^m}{F_{n}^m}}.

What is the limit of the generalized golden ratio below? limmϕm=limmlimnFn+1mFnm\lim_{m \rightarrow \infty}{\phi_m} = \lim_{m \rightarrow \infty}{\lim_{n \rightarrow \infty}{\frac{F_{n+1}^m}{F_{n}^m}}}

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