My Pi Discovery(Reason why 22/7 works)

[greggory]

So, I was working with the Fibonacci sequence this afternoon, and I stumbled onto something that maybe a major discovery to geometry, I don't know if it is or not. So, I began working with the golden spiral, and began drawing it. After drawing it, I noticed you could draw circles and more circles with this sequence and spiral. The values even made sense to the circle sketches.

This is what I did. Here is an image of it:

http://img442.imageshack.us/img442/7633/proof1oa.png

What I did was took three numbers and inputted them into the formula I made, r1 / D + r2 / D = pi/2. The first numbers I did this to, I simply just got 2/2. Now, as I went on, the numbers became closer and closer to pi. I realized maybe if I kept going, I would reach a value, which I did. What I found maybe the reason why 22/7 works as an approximation of pi. So, I soon ended up with the diameter as 21. I did 1+1+2+3+5+8 / 21 + 13 / 21 = 3.14/2. Then, this would be 20/21 + 13 / 21. Add these fractions together, and you get 33/21, which simplified would be 11/7. If you were to multiply this by 2, you would get 22/7.

I was just wondering if this work is legit or not.

 

[Char. Limit]

Unfortunately, the only reason that this seems to work is that phi, or 1.618, approximately, is somewhat close to pi/2. You're right that your equation is converging, but wrong in what it's converging to. It's not converging on pi/2, but on phi. Now, let's take a look at this to see why it's true...

Let's put your denominator as the nth Fibonacci number, F(n). If this is the case, then b will be F(n-1), and a will be the sum of all Fibonacci numbers up to F(n-2). Now, it is the case that the sum of all Fibonacci numbers up to F(n-2) is F(n)-1, or in summation notation:

$$\sum_{k=1}^{n-2} F(k) = F(n)-1$$

By substituting these in, we get that the left side is equal to:

$$\frac{F(n)+F(n-1)-1}{F(n)} = 1 + \frac{F(n-1)}{F(n)} - \frac{1}{F(n)}$$

Expanding.Now, the first term is obviously 1, and the third term, as n goes large, will tend to zero. The second term is the most interesting, however. It's rather well known that the ratio of successive Fibonacci numbers tends to phi, i.e. F(n)/F(n-1) tends to phi for large n. This means that F(n-1)/F(n) tends to 1/phi for large n, and 1/phi can be written as phi-1 (it's one of the properties of phi). Substituting all this in, we get the following result:

$$\lim_{n\to\infty} \frac{F(n)+F(n-1)-1}{F(n)} = \phi$$

Close to pi/2? Certainly. Converging to pi/2? Not quite.