the Fibonacci sequence and series limits

[Zardi]

Let Fn denote the Fibonacci sequence, i.e., F1 = 1, F2 = 1 and Fn+1 = Fn + Fn-1
for n = 2;3; : : : . Let an = Fn+1/Fn. Show that if an converges then the limit must be
(1 + 5^1/2)/2.

 

[stapel]

Let Fn denote the Fibonacci sequence, i.e., F1 = 1, F2 = 1 and Fn+1 = Fn + Fn-1
for n = 2;3; : : : . Let an = Fn+1/Fn. Show that if an converges then the limit must be
(1 + 5^1/2)/2.

There are various proofs archived online, such as here. Did you have a specific question about a specific proof, or are you asking something else? Please be specific. Thank you! ;-)