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April 24th, 2015, 07:00 AM  #1 
Newbie Joined: Apr 2015 From: washington Posts: 1 Thanks: 0  FibonacciType 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(n1,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^(n2), G(k+1,n) = (G(k,n) + G(k1,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(n1,n) > sqrt(e), Or equivalently, ((1 + sqrt(1 + 4*n^2))/(2*n))^(2*n) > sqrt(e) This convergence is much faster. 
April 24th, 2015, 08:02 AM  #2 
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms 
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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fibonaccitype, number, number e, series 
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