Difference between revisions of "Limit of quotient of consecutive Fibonacci numbers"
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==Theorem== | ==Theorem== | ||
The following formula holds: | The following formula holds: | ||
| − | $$\displaystyle\lim_{n \rightarrow \infty} \dfrac{ | + | $$\displaystyle\lim_{n \rightarrow \infty} \dfrac{F(n+1)}{F(n)}=\varphi,$$ |
| − | where $ | + | where $F(n)$ denotes the $n$th [[Fibonacci numbers|Fibonacci number]] and $\varphi$ denotes the [[golden ratio]]. |
==Proof== | ==Proof== | ||
==References== | ==References== | ||
| − | * {{PaperReference|Sur la série des inverse de nombres de Fibonacci|1899|Edmund Landau|prev=Fibonacci | + | * {{PaperReference|Sur la série des inverse de nombres de Fibonacci|1899|Edmund Landau|prev=Fibonacci numbers|next=Reciprocal Fibonacci constant}} |
[[Category:Theorem]] | [[Category:Theorem]] | ||
[[Category:Unproven]] | [[Category:Unproven]] | ||
Latest revision as of 23:53, 6 June 2017
Theorem
The following formula holds: $$\displaystyle\lim_{n \rightarrow \infty} \dfrac{F(n+1)}{F(n)}=\varphi,$$ where $F(n)$ denotes the $n$th Fibonacci number and $\varphi$ denotes the golden ratio.
