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

%I

%S 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,

%T 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,

%U 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

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

%C 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.

%C 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.

%C Yes, this number is transcendental. It is among various such forms Kempner showed are transcendental. - _Kevin Ryde_, Jul 12 2019

%H B. Adamczewski, <a href="https://arxiv.org/abs/1303.1685">The Many Faces of the Kempner Number</a>, arXiv:1303.1685 [math.NT], 2013.

%H Aubrey J. Kempner, <a href="http://www.jstor.org/stable/1988833">On Transcendental Numbers</a>, Transactions of the American Mathematical Society 17 (1916), pp. 476-482.

%H Kurt Mahler, <a href="http://dx.doi.org/10.1007/BF01194652">Arithmetische Eigenschaften einer Klasse transzendental-transzendenter Funktionen</a>, Math. Z. 32 (1930), 545-585.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

%e 0.308609008556231856400340479718025221697433904166441366801367221...

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

%Y Cf. A030300 (binary expansion), A160386.

%Y Cf. A007404, A036987, A275973.

%K nonn,cons

%O 0,1

%A _Stanislav Sykora_, Aug 15 2016

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Last modified August 6 14:33 EDT 2020. Contains 336246 sequences. (Running on oeis4.)