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%I #21 Nov 02 2014 12:18:36
%S 1,3,9,8,5,9,8,5,1,6,3,2,9,3,7,8,7,1,8,7,8,5,9,0,5,5,5,2,8,1,7,6,1,4,
%T 1,8,5,5,6,0,3,3,6,5,4,0,9,8,8,4,6,3,9,3,6,4,5,0,0,9,2,0,2,4,8,8,5,5,
%U 6,0,9,7,1,6,4,7,0,4,8,1,5,5,6,8,8,1
%N Decimal expansion of continued fraction [x(1),x(2),x(3),...], where x(n) = F(n+1)/F(n), where F = A000045 (Fibonacci numbers).
%C Suppose that x(n) is a sequence of positive real numbers with divergent sum. By the Seidel Convergence Theorem, the continued fraction [x(1),x(2),x(3),...] converges.
%C The fact that F(n+1)/F(n) is the n-th convergent of a continued fraction (specifically, of the golden ratio) exemplifies a certain function f of a positive real variable x: let p(i)/q(i), for i >=0, be the convergents to x; then f(x) is the number [p(0)/q(0), p(1)/q(1), p(2)/q(2), ... ]. For x = golden ratio, f(x) = 1.398598..., f(f(x)) = 1.4903397..., f(f(f(x))) = 1.4995061.... Let L(x) = lim(f(n,x)), where f(0,x) = x, f(1,x) = f(x), and f(n,x) = f(f(n-1,x)). It appears that L(golden ratio) = 3/2.
%F [1, [1,1], [1,1,1], [1,1,1,1], ... ]. (Here, as in the Name and Example sections, square brackets indicate continued fractions.)
%e [1, 2/1, 3/2, 5/3, 8/5,...] = [1,2,1,1,27,1,16,670,9,3,2,1,13,1,4,1,1,1...] = 1.3985985... The first 5 ordinary convergents are 1, 3/2, 4/3, 7/5, 193/138.
%t z = 500; t = Table[Fibonacci[n + 1]/Fibonacci[n], {n, z}]
%t r = FromContinuedFraction[t]; c = Convergents[r, z];
%t Numerator[c] (* A229348 *)
%t Denominator[c] (* A229349 *)
%t RealDigits[r, 10, 120] (* A229350 *)
%Y Cf. A229348, A229349, A229351.
%K nonn,cons,easy
%O 1,2
%A _Clark Kimberling_, Sep 21 2013
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