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A096444 Decimal expansion of (Pi - 1)/2.

%I A096444 #65 Sep 08 2022 08:45:14
%S A096444 1,0,7,0,7,9,6,3,2,6,7,9,4,8,9,6,6,1,9,2,3,1,3,2,1,6,9,1,6,3,9,7,5,1,
%T A096444 4,4,2,0,9,8,5,8,4,6,9,9,6,8,7,5,5,2,9,1,0,4,8,7,4,7,2,2,9,6,1,5,3,9,
%U A096444 0,8,2,0,3,1,4,3,1,0,4,4,9,9,3,1,4,0,1,7,4,1,2,6,7,1,0,5,8,5,3,3,9,9,1,0,7
%N A096444 Decimal expansion of (Pi - 1)/2.
%C A096444 From _Bernard Schott_, Apr 19 2021: (Start)
%C A096444 The series Sum_{k>=1} sin(k)/k and also Sum_{k>=1} cos(k)/k (A121225) are called Fresnel series.
%C A096444 The series Sum_{k>=1} |sin(k)/k| is divergent. (End)
%D A096444 Xavier Merlin, Methodix Analyse, Ellipses, 1997, p. 117.
%H A096444 Ira Rosenholtz, <a href="http://www.jstor.org/stable/2690793">Tangent sequences, world records, Pi, and the meaning of life: Some Applications of Number Theory to Calculus</a>, Math. Mag., Vol. 72, No. 5 (1999), pp. 367-376.
%H A096444 Jan W. H. Swanepoel, <a href="http://dx.doi.org/10.1016/j.jnt.2014.10.002">On a generalization of a theorem by Euler</a>, Journal of Number Theory, Vol. 149 (April 2015), pp. 46-56.
%H A096444 Université de Rennes, <a href="http://braise.univ-rennes1.fr/donnees/ParamHTML/S%E9ries%20num%E9riques/con/S%E9ries%20semi-convergentes/cst.pdf">Séries semi-convergentes</a>, Base raisonnée d'exercices de Mathématiques, p. 2 (Braise).
%H A096444 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A096444 Equals Sum_{k >= 1} sin(k)/k. (This follows from the identity x = Pi - 2 Sum_{k >= 1} sin(k*x)/k, as observed by Euler in 1744.)
%F A096444 Equals A019669 minus 1/2. - _R. J. Mathar_, Dec 15 2008
%F A096444 Equals Sum_{k >= 1} (sin(k)/k)^2. (Interestingly, Sum_{k >= 1} sin(k)/k = Sum_{k >= 1} (sin(k)/k)^2, a series whose terms sum to the sum of the square of each term.) - _Dimitri Papadopoulos_, Mar 11 2015
%F A096444 Equals arctan(sin(1)/(1-cos(1))). - _Amiram Eldar_, Jun 06 2021
%e A096444 1.0707963267948966...
%t A096444 RealDigits[(Pi - 1)/2, 10, 105][[1]] (* _Robert G. Wilson v_, Aug 26 2004 *)
%o A096444 (PARI) (Pi-1)/2 \\ _Charles R Greathouse IV_, Jan 30 2015
%o A096444 (Magma) pi:=Pi(RealField(110)); Reverse(Intseq(Floor(10^105*(pi-1)/2))); // _Vincenzo Librandi_, Mar 12 2015
%Y A096444 Cf. A019669, A096418, A121225, A342680.
%K A096444 nonn,cons
%O A096444 1,3
%A A096444 _N. J. A. Sloane_, Aug 16 2004
%E A096444 More terms from _Robert G. Wilson v_, Aug 17 2004
%E A096444 Better definition from _Eric W. Weisstein_, Aug 18 2004