Fibonacci limit

Calculus Level 1

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

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

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

×

Problem Loading...

Note Loading...

Set Loading...