My Math Forum Fibonacci-Type series and the number e

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 April 24th, 2015, 07:00 AM #1 Newbie   Joined: Apr 2015 From: washington Posts: 1 Thanks: 0 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.
 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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