OFFSET
1,2
COMMENTS
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.
EXAMPLE
[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.
MATHEMATICA
z = 500; t = Table[(n+1)/n, {n, z}]
r = FromContinuedFraction[t]; c = Convergents[r, z];
Numerator[c] (* A229351 *)
Denominator[c] (* A229352 *)
RealDigits[r, 10, 120] (* A229353 *)
Take[Convergents[FromContinuedFraction[Table[(n+1)/n, {n, 100}]]]//Denominator, 30] (* Harvey P. Dale, Jan 31 2023 *)
CROSSREFS
KEYWORD
nonn,frac,easy
AUTHOR
Clark Kimberling, Sep 21 2013
STATUS
approved