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%I #9 Jan 31 2023 16:49:03
%S 1,2,65,67,132,199,1923,9814,11737,21551,54839,76390,665959,1408308,
%T 3482575,11856033,133898938,547451785,681350723,1228802508,1910153231,
%U 3138955739,8188064709,19515085157,125278575651,144793660808,414865897267,559659558075
%N Denominators 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 *)
%t Take[Convergents[FromContinuedFraction[Table[(n+1)/n,{n,100}]]]//Denominator,30] (* _Harvey P. Dale_, Jan 31 2023 *)
%Y Cf. A229348, A229351, A229353.
%K nonn,frac,easy
%O 1,2
%A _Clark Kimberling_, Sep 21 2013
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