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A332890 Decimal expansion of Sum_{k>=0} 1/(4*k)!.

%I A332890 #53 Apr 05 2024 08:17:46
%S A332890 1,0,4,1,6,9,1,4,7,0,3,4,1,6,9,1,7,4,7,9,3,9,4,2,1,1,1,4,1,0,0,0,1,9,
%T A332890 1,4,3,1,6,6,9,1,9,7,6,6,4,9,1,8,9,2,9,6,6,2,0,3,7,4,9,7,3,5,0,4,5,9,
%U A332890 3,4,7,2,8,9,1,1,8,4,7,7,3,1,7,4,1,1,0
%N A332890 Decimal expansion of Sum_{k>=0} 1/(4*k)!.
%C A332890 For q integer >= 1, Sum_{m>=0} 1/(q*m)! = (1/q) * Sum_{k=1..q} exp(X_k) where X_1, X_2, ..., X_q are the q-th roots of unity.
%D A332890 Serge Francinou, Hervé Gianella, Serge Nicolas, Exercices de Mathématiques, Oraux X-ENS, Analyse 2, problème 3.10 p. 182, Cassini, Paris, 2004.
%H A332890 Michael I. Shamos, <a href="http://euro.ecom.cmu.edu/people/faculty/mshamos/cat.pdf">A catalog of the real numbers</a>, (2011). See p. 76.
%F A332890 Equals (1/2) * (cos(1) + cosh(1)).
%F A332890 Equals (1/2) * Sum_{k>=0} (1 + (-1)^k)/((2*k)!). - _Peter Luschny_, Mar 01 2020
%F A332890 Sum_{k>=0} (-1)^k / (4*k)! = cos(1/sqrt(2)) * cosh(1/sqrt(2)) = 0.958358132833... - _Vaclav Kotesovec_, Mar 02 2020
%F A332890 Continued fraction: 1 + 1/(24 - 24/(1681 - 1680/(11881 - ... - P(n-1)/((P(n) + 1) - ... )))), where P(n) = (4*n)*(4*n - 1)*(4*n - 2)*(4*n - 3) for n >= 1. Cf. A346441. - _Peter Bala_, Feb 22 2024
%e A332890 1.0416914703416917479394211141000191431669197664918929...
%p A332890 evalf(1/2 * (cos(1) + cosh(1)), 100);
%t A332890 RealDigits[Sum[1/(4n)!,{n,0,\[Infinity]}],10,120][[1]] (* _Harvey P. Dale_, Apr 18 2023 *)
%o A332890 (PARI) suminf(k=0,(1 + (-1)^k)/((2*k)!))/2 \\ _Hugo Pfoertner_, Mar 01 2020
%o A332890 (PARI) suminf(k=0, 1/(4*k)!) \\ _Michel Marcus_, Mar 02 2020
%Y A332890 Cf. A001113 (Sum 1/k!), A073743 (Sum 1/(2k)!),  A143819 (Sum 1/(3k)!), this sequence (Sum 1/(4k)!), A269296 (Sum 1/(5k)!), A332892 (Sum 1/(6k)!), A346441.
%K A332890 nonn,cons
%O A332890 1,3
%A A332890 _Bernard Schott_, Mar 01 2020
%E A332890 More terms from _Hugo Pfoertner_, Mar 02 2020