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

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, 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, 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

Offset

1,3

Comments

From Bernard Schott, Apr 19 2021: (Start)

The series Sum_{k>=1} sin(k)/k and also Sum_{k>=1} cos(k)/k (A121225) are called Fresnel series.

The series Sum_{k>=1} |sin(k)/k| is divergent. (End)

References

Xavier Merlin, Methodix Analyse, Ellipses, 1997, p. 117.

Links

Table of n, a(n) for n=1..105.

Ira Rosenholtz, Tangent sequences, world records, Pi, and the meaning of life: Some Applications of Number Theory to Calculus, Math. Mag., Vol. 72, No. 5 (1999), pp. 367-376.

Jan W. H. Swanepoel, On a generalization of a theorem by Euler, Journal of Number Theory, Vol. 149 (April 2015), pp. 46-56.

Université de Rennes, Séries semi-convergentes, Base raisonnée d'exercices de Mathématiques, p. 2 (Braise).

Index entries for transcendental numbers.

Formula

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.)

Equals A019669 minus 1/2. - R. J. Mathar, Dec 15 2008

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

Equals arctan(sin(1)/(1-cos(1))). - Amiram Eldar, Jun 06 2021

Example

1.0707963267948966...

Mathematica

RealDigits[(Pi - 1)/2, 10, 105][[1]] (* Robert G. Wilson v, Aug 26 2004 *)

Prog

(PARI) (Pi-1)/2 \\ Charles R Greathouse IV, Jan 30 2015
(Magma) pi:=Pi(RealField(110)); Reverse(Intseq(Floor(10^105*(pi-1)/2))); // Vincenzo Librandi, Mar 12 2015

Crossrefs

Cf. A019669, A096418, A121225, A342680.

Sequence in context: A216185 A202996 A019597 * A246918 A055957 A165090

Adjacent sequences: A096441 A096442 A096443 * A096445 A096446 A096447

Keyword

nonn,cons

Author

N. J. A. Sloane, Aug 16 2004

Extensions

More terms from Robert G. Wilson v, Aug 17 2004

Better definition from Eric W. Weisstein, Aug 18 2004

Status

approved