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

Author

Stanislav Sykora, Aug 15 2016

Keywords

• nonn
• cons

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

Examples

			0.308609008556231856400340479718025221697433904166441366801367221...
		

Links

• 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.

Crossrefs

Cf. A030300 (binary expansion), A160386.

Cf. A007404, A036987, A275973.

Programs

Mathematica

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

PARI

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