A229987 Decimal expansion of the upper limit of the convergents of the continued fraction [1, -1/2, 1/4, -1/8, ... ].

1, 4, 0, 1, 2, 8, 4, 1, 1, 9, 2, 2, 8, 4, 2, 9, 8, 8, 4, 2, 5, 9, 6, 2, 5, 6, 6, 9, 8, 7, 9, 6, 5, 5, 0, 4, 0, 9, 1, 1, 8, 7, 4, 2, 4, 8, 7, 5, 4, 8, 2, 4, 1, 1, 2, 4, 1, 8, 1, 8, 5, 5, 5, 7, 3, 7, 5, 0, 0, 5, 2, 6, 8, 1, 6, 8, 3, 3, 2, 4, 0, 1, 5, 1, 1, 3

Offset

1,2

Comments

Since sum{(-1)*2^(-k), k = 0,1,2,...} converges, the convergents of [1, -1/2, 1/4, -1/8, ... ] diverge, by the Seidel Convergence Theorem. However, the odd-numbered convergents converge, as do the even-numbered convergents. In the Example section, these limits are denoted by u and v.

Links

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

Example

u = 1.401284... = [1, 2, 2, 30, 1, 2, 1, 254, 1, 6, 1, 2046, 1, ...];
v = -0.48360... = [0, -2, -14, -1, -2, -1, -126, -1, -6, -1,  ...].
Every term of the continued fraction of u is 1 or has the form -2 + 2^m; every term for v is -1 or has the form 2 - 2^m.

Mathematica

$MaxExtraPrecision = Infinity; z = 800; t = Table[((-1)^n) 2^(-n), {n, 0, z}];  u = N[Convergents[t][[z - 1]], 120]; v = N[Convergents[t][[z]], 120]; RealDigits[u] (* A229987 *)
RealDigits[v] (* A229988 *)

Crossrefs

Cf. A229988.

Sequence in context: A365094 A179296 A196817 * A343953 A307769 A096793

Adjacent sequences: A229984 A229985 A229986 * A229988 A229989 A229990

Keyword

nonn,cons

Author

Clark Kimberling, Oct 06 2013

Status

approved