Difference between revisions of "Limit of quotient of consecutive Fibonacci numbers"

From specialfunctionswiki
Jump to: navigation, search
 
(One intermediate revision by the same user not shown)
Line 1: Line 1:
 
==Theorem==
 
==Theorem==
 
The following formula holds:
 
The following formula holds:
$$\displaystyle\lim_{n \rightarrow \infty} \dfrac{F_{n+1}}{F_n}=\phi,$$
+
$$\displaystyle\lim_{n \rightarrow \infty} \dfrac{F(n+1)}{F(n)}=\varphi,$$
where $F_n$ denotes the $n$th [[Fibonacci numbers|Fibonacci number]] and $\phi$ denotes the [[golden ratio]].
+
where $F(n)$ denotes the $n$th [[Fibonacci numbers|Fibonacci number]] and $\varphi$ denotes the [[golden ratio]].
  
 
==Proof==
 
==Proof==

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.

Proof

References