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April 24th, 2015, 07:00 AM   #1
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Fibonacci-Type series and the number e

Recently, I proved what I think are amazing convergence formulas that show a connection between the golden ratio, the fibonacci series, and e = 2.718281828..., the base of the natural logarithms. However, I have not seen these proofs anywhere in the literature. I'd greatly appreciate any comment on any of these or similar results.

The first result is the following, as n grows to infinity:

G(n-1,n) + G(n,n+1) ----> sqrt(e)

Where, for a given n, the series G(k,n) is defined as:

G(0,n) = 0, G(1,n) = n^(n-2), G(k+1,n) = (G(k,n) + G(k-1,n))/n^2

Clearly, G(k, 1) is just the standard Fibonacci series. I also proved that:

G(n,n)*(1 + sqrt(1 + 4*n^2))/2 + G(n-1,n) ----> sqrt(e),

Or equivalently,

((1 + sqrt(1 + 4*n^2))/(2*n))^(2*n) ---> sqrt(e)

This convergence is much faster.
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April 24th, 2015, 08:02 AM   #2
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Is the function G known? It seems like a fairly complicated definition, so unless I'm missing something I wouldn't expect that someone would have written about that particular function.
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