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
Examples
1.0707963267948966...
References
- Xavier Merlin, Methodix Analyse, Ellipses, 1997, p. 117.
Links
- 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.
Programs
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Magma
pi:=Pi(RealField(110)); Reverse(Intseq(Floor(10^105*(pi-1)/2))); // Vincenzo Librandi, Mar 12 2015
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Mathematica
RealDigits[(Pi - 1)/2, 10, 105][[1]] (* Robert G. Wilson v, Aug 26 2004 *)
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PARI
(Pi-1)/2 \\ Charles R Greathouse IV, Jan 30 2015
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
Extensions
More terms from Robert G. Wilson v, Aug 17 2004
Better definition from Eric W. Weisstein, Aug 18 2004
Comments