A275975 Decimal expansion of Sum_{k>=0}((-1)^k/2^(2^k)).

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

Offset

0,1

Comments

Except for the alternating signs, this constant is defined in a similar way to the Kempner-Mahler number A007404. It is related to the Jeffreys binary sequence A275973 somewhat like Kempner-Mahler number is related to the Fredholm-Rueppel sequence A036987.

Conjecture: Numbers of the type Sum_{k>=0}(x^(2^k)) with algebraic x and |x|<1 are known to be transcendental (Mahler 1930, Adamczewski 2013). It is likely that the alternating sign does not invalidate this property.

Yes, this number is transcendental. It is among various such forms Kempner showed are transcendental. - Kevin Ryde, Jul 12 2019

Links

Table of n, a(n) for n=0..104.

Boris Adamczewski, The Many Faces of the Kempner Number, arXiv:1303.1685 [math.NT], 2013.

Aubrey J. Kempner, On Transcendental Numbers, Transactions of the American Mathematical Society 17 (1916), pp. 476-482.

Kurt Mahler, Arithmetische Eigenschaften einer Klasse transzendental-transzendenter Funktionen, Math. Z. 32 (1930), 545-585.

Index entries for transcendental numbers.

Example

0.308609008556231856400340479718025221697433904166441366801367221...

Mathematica

RealDigits[N[Sum[((-1)^k/2^(2^k)), {k, 0, Infinity}], 120]][[1]] (* Amiram Eldar, Jun 11 2023 *)

Prog

(PARI) default(realprecision, 2100); suminf(k=0, (-1)^k*0.5^2^k)

Crossrefs

Cf. A030300 (binary expansion), A160386.

Cf. A007404, A036987, A275973.

Sequence in context: A333567 A248424 A292525 * A201665 A137204 A021328

Adjacent sequences: A275972 A275973 A275974 * A275976 A275977 A275978

Keyword

nonn,cons

Author

Stanislav Sykora, Aug 15 2016

Status

approved