I'm back. This time, however, I need a little help.

I'm trying to come up with a way to determine an implicitly defined function for a sequence of ratios that arise from sequences related to the Fibonacci sequence. (I can't seem to come up with a good synonym for "sequence".)

Let me elaborate. The first sequence can be defined as such: For any number in the sequence, F(n)=F(n-1). The second sequence is: F(n) = F(n-1) + F(n-2). The third sequence is F(n) = F(n-1) + F(n-2) + F(n-3). So, the jth sequence is F(n) = Sum(j, k=1) [F(n-k)].

The bigger implicitly defined function spits out ratios that arise from these sequences. For each sequence, Lim(n-->∞) [F(n)/F(n-1)] is a constant. You may have recognized the second sequence as the Fibonacci sequence. The limit of F(n)/F(n-1) as n approaches infinity for the Fibonacci sequence is phi- the golden ratio. This result can easily be proven.

First, we assume that the limit must converge on some constant- x. So,

x = Lim(n-->∞) [F(n)/F(n-1)].

Then, we substitute for F(n)'s definition.

x = Lim(n-->∞) [{F(n-1) + F(n-2)}/F(n-1)]

Simplifying,

x = Lim(n-->∞) [1+{F(n-2)/F(n-1)}]

Since Lim(n-->∞) [F(n-2)/F(n-1)] = Lim(n-->∞) [F(n-1)/F(n)],

x = 1 + Lim(n-->∞) [F(n-1)/F(n)].

Since Lim(n-->∞) [F(n-1)/F(n)] = 1/x,

x = 1 + 1/x --> x^2 = x + 1 --> x^2 - x - 1 = 0 and x = (1 + rt[5])/2. (The negative result is excluded since the entire set is positive, and therefore a ratio of any two numbers in the set must also be positive.)

It is easy for this number of terms, but for a sequence with 14 terms the limit cannot be simplified as easily. (I only got as far as 4.) However, I came up with a slightly more easy way to find the ratio.

R(n) = The positive solution to x^n = Sum({n-1}, k=0) [x^k].

This may look ugly, but in fact I find it quite pretty. For the 3rd sequence, the ratio is the positive solution to x^3 = x^2 + x + 1. (Around 1.839)

I've noticed an interesting property of the ratios for the sequences. I had my computer calculate all the ratios up to the 10th one, and as far as I can see the limit of the ratios is 2. So, Lim(n-->∞) [R(n)] = 2. (The 1oth ratio is very close to 2- 1.999018633.)

The problem I'm having is writing the function R(n) implicitly. This is because there's no way to factor x^n - x^(n-1) - x^(n-2) - ... - 1 = 0 that I can come up with, and I'm pretty sure it can't be done using elementary algebraic techniques. My instinct at this point is to gather some other properties of the graph and form some kind of estimate function for R(n). Experts on Taylor Series?