Why is the Fibonacci ratio though a decreasing function, it is alternating and decreasing? 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) + '=&gt;' + 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.

Thanks
Consider the formula you used for calculating the values: $$F_n = \frac{1}{\sqrt{5}}\left(\phi^n - \psi^n\right)$$ where $\phi=\frac{1}{2}\left(1+\sqrt{5}\right)$ and $\psi=\frac{1}{2}\left(1-\sqrt{5}\right)$.

The ratio of consecutive terms can then be expressed as $$\frac{F_{n+1}}{F_n}=\frac{\phi^{n+1}-\psi^{n+1}}{\phi^n - \psi^n} = \phi + \frac{(\phi-\psi)\psi^n}{\phi^n -\psi^n}$$

There are is a couple of observations we can now make:


We have $|\phi|&gt;1$ and $|\psi|&lt;1$. Thus, with increasing $n$, the value of $\psi^n$ will get closer and closer to zero, while $\phi^n$ will grow further and further. This already implies the ratio $(F_{n+1}/F_n)$ converges to $\phi$, since the other term converges to zero (its numerator is going to zero and denominator to infinity).
$\psi&lt;0$, so $\psi^n$ flips its sign whenever we increase $n$ by one. Therefore, the second term oscillates from positives to negatives; making the ratio alternate between being greater than $\phi$ and smaller than $\phi$.


The first observation can be generalized to any sequence which can be expressed as a linear combination of exponentials, the greatest of which has base greater (in absolute value) than 1. The ratio of consecutive terms will converge to this greatest base.
Let $F_i$ be the $i$-th Fibonacci number. Put $F_i$ people at one table, and deliver $F_{i-1}$ pizzas to them. Put $F_{i+1}$ people at a second table and deliver $F_i$ pizzas to them. At the first table, there are $F_{i-1}/F_i$ pizzas per person, and at the second table there are $F_i/F_{i+1}$ pizzas per person.

Now have everyone at the first table move to the second table, and bring their pizzas along. In total, there are $F_i + F_{i+1} = F_{i+2}$ people and $F_{i-1} + F_i = F_{i+1}$ pizzas, so we have $F_{i+1}/F_{i+2}$ pizzas per person. If there were more pizzas per person at the first table than at the second (in other words, if $F_{i-1}/F_i &gt; F_i/F_{i+1}$), then bringing over the people from the first table increased the number of pizzas per person at the second table. In other words, $F_{i+1}/F_{i+2} &gt; F_i/F_{i+1}$. If there were more pizzas per person at the second table ($F_{i-1}/F_i &lt; F_i/F_{i+1}$), then bringing over the people from the first table decreased the number of pizzas per person, and $F_{i+1}/F_{i+2} &lt; F_i/F_{i+1}$. This shows the oscillation. Note that either way, $F_{i+1}/F_{i+2}$ sits in between $F_{i-1}/F_i$ and $F_i/F_{i+1}$. The sequence of ratios converges by the alternating series test.

I phrased this in terms of ratios smaller than $1$ to avoid the appearance of gluttony, but obviously you can do the same thing with ratios greater than $1$ ($F_{i+1}/F_i$), or just take reciprocals.