I tried to find the ratio of consecutive terms of the Fibonacci series and found that it is a decreasing function and it converges . I tried it though a small code piece in python so that I can have a lot of data points to analyze.
import math def F(n): return ((1+math.sqrt(5))**n-(1-math.sqrt(5))**n)/(2**n*math.sqrt(5)) for i in range(53): a = F(i) b = F(i+1) print str(b) + '/' + str(a) + '=>' + str(b/a)
And it gave values which suggested that it is a decreasing function and it converges at around 1.618. But I found that, though it was decreasing, it was not a continuously decreasing function, it in fact increases and decreases alternatively. But overall it was decreasing and converging.
And I tried to plot the values to prove this graphically.
(I expanded the graph a bit by plotting for $n*3$, just to make the view clear)
My doubt is this - What is so special in this series which makes it behave alternating in nature?
I tried to do it for consecutive numbers ratio and found that it was a increasing but not alternating like the Fibonacci ratios.