A229351 Numerators of the ordinary convergents of continued fraction [2/1, 3/2, 4/3, 5/4,...].

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, 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, 2, 1, 6, 1, 9, 2, 9, 2, 6, 8, 2, 3, 8, 8, 6, 9, 5, 2

Offset

1,1

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.

Links

Table of n, a(n) for n=1..86.

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 *)

Crossrefs

Cf. A229348, A229352, A229353.

Sequence in context: A340884 A011181 A134822 * A229353 A133057 A155523

Adjacent sequences: A229348 A229349 A229350 * A229352 A229353 A229354

Keyword

nonn,frac,easy

Author

Clark Kimberling, Sep 21 2013

Status

approved