%I
%S 2,4,9,2,4,5,9,7,4,6,0,2,1,2,8,6,6,0,3,3,9,6,8,5,1,8,3,2,3,9,1,5,0,8,
%T 5,2,2,6,6,0,6,4,3,8,9,0,5,2,9,8,4,8,0,2,5,5,5,3,3,5,2,3,5,8,0,0,6,2,
%U 2,1,6,1,9,2,9,2,6,8,2,3,8,8,6,9,5,2
%N Numerators of the ordinary convergents of continued fraction [2/1, 3/2, 4/3, 5/4,...].
%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.
%e [2/1, 3/2, 4/3, 5/4, ...] = 2.492459746021286... The first 5 ordinary convergents are 2, 5/2, 162/65, 167/67, 329/132.
%t z = 500; t = Table[(n+1)/n, {n, z}]
%t r = FromContinuedFraction[t]; c = Convergents[r, z];
%t Numerator[c] (* A229351 *)
%t Denominator[c] (* A229352 *)
%t RealDigits[r, 10, 120] (* A229353 *)
%Y Cf. A229348, A229352, A229353.
%K nonn,frac,easy
%O 1,1
%A _Clark Kimberling_, Sep 21 2013
