A342680 -id:A342680 - cp's OEIS frontend

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

N. J. A. Sloane, Aug 16 2004

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)

			1.0707963267948966...

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

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.

Cf. A019669, A096418, A121225, A342680.

pi:=Pi(RealField(110)); Reverse(Intseq(Floor(10^105*(pi-1)/2))); // Vincenzo Librandi, Mar 12 2015

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

(Pi-1)/2 \\ Charles R Greathouse IV, Jan 30 2015

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

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

Better definition from Eric W. Weisstein, Aug 18 2004

A362662 Decimal expansion of Sum_{n>=1} (tan(1/n) - sin(1/n)).

Original entry on oeis.org

8, 2, 2, 0, 8, 2, 2, 0, 0, 8, 0, 3, 5, 8, 8, 2, 0, 2, 9, 3, 5, 8, 7, 0, 1, 1, 8, 7, 1, 5, 9, 9, 3, 5, 2, 0, 7, 3, 0, 4, 4, 6, 0, 4, 3, 8, 1, 1, 6, 5, 3, 2, 6, 3, 9, 0, 8, 3, 6, 8, 5, 9, 3, 9, 3, 4, 3, 7, 1, 0, 5, 3, 4, 5, 3, 5, 4, 3, 6, 8, 1, 3, 2, 4, 6, 0, 0, 4, 7, 1, 3, 4, 7, 4, 3, 2, 2

Offset: 0

Bernard Schott, Apr 29 2023

Series Sum_{n>=1} sin(1/n) and Sum_{n>=1} tan(1/n) -> oo but with u(n) = (tan(1/n) - sin(1/n)), as u(n) ~ 1 / (2*n^3) when n -> oo, the series Sum_{n>=1} u(n) is convergent.

			Equals 0.822082200803588202935870118715993520730...

J. Guégand and M.-A. Maingueneau, Exercices d'Analyse, Exercice 1 - 41.2, p. 47, Classes Préparatoires aux Grandes Ecoles, Ellipses, 1988.

Cf. A096444, A248946, A248948, A342680.

evalf(sum(tan(1/n) - sin(1/n), n=1..infinity), 120);

sumpos(n=1, tan(1/n) - sin(1/n)) \\ Michel Marcus, Apr 29 2023

A351738 Decimal expansion of Sum_{k>0} sin(sqrt(k)) / k.

Original entry on oeis.org

1, 7, 1, 5, 6, 7, 1, 7, 9, 4, 7, 0, 9

Offset: 1

Bernard Schott, May 20 2022

Sum_{k>0} sin(k^alpha) / (k^beta) with 0 < alpha < 1 is convergent if beta > max(alpha, 1-alpha); the constant of this sequence corresponds to the case alpha = 1/2 and beta = 1 (see Arnaudiès).
Consequence: Sum_{k>0} sin(k^(1/m)) / k converges for any positive integer m.
The sequence converges slowly.

			1.715671794709...

J. M. Arnaudiès, P. Delezoide et H. Fraysse, Exercices résolus d'Analyse du cours de mathématiques - 2, Dunod, 1993, Exercice 11, pp. 316-319.

Mathematics Stack Exchange, Convergence of Sum_{k=0..infinity} sin(sqrt(k)) / k.

Cf. A096418, A096444, A114940, A228639, A263193, A342680.

default(realprecision, 100); sumalt(k=0, sum(j=1+floor(k^2*Pi^2),floor((k+1)^2*Pi^2), sin(sqrt(j))/j)) \\ Vaclav Kotesovec, May 21 2022

More digits from Stefano Spezia, May 21 2022

A363906 Decimal expansion of Sum_{n>=1} (arcsin(1/n) - sin(1/n)).

Original entry on oeis.org

7, 9, 9, 5, 8, 8, 6, 2, 3, 5, 5, 3, 3, 7, 6, 9, 9, 0, 1, 1, 3, 9, 9, 1, 1, 1, 3, 5, 2, 7, 2, 3, 9, 8, 2, 5, 0, 4, 0, 1, 7, 2, 2, 8, 4, 1, 9, 0, 7, 7, 7, 9, 6, 8, 3, 6, 4, 1, 1, 6, 5, 9, 2, 8, 4, 3, 6, 7, 7, 3, 0, 4, 0, 6, 7, 7, 5, 5, 7, 2, 1, 7, 9, 1, 8, 1, 7

Offset: 0

Bernard Schott, Jun 27 2023

Series Sum_{n>=1} arcsin(1/n) and Sum_{n>=1} sin(1/n) -> oo but with v(n) = (arcsin(1/n) - sin(1/n)), as v(n) ~ 1 / (3*n^3) when n -> oo, the series Sum_{n>=1} v(n) is convergent.

			0.79958862355337699...

Cf. A096444, A248946, A342680, A362662.

NSum[ArcSin[1/n]-Sin[1/n], {n, Infinity}, WorkingPrecision -> 95, NSumTerms -> 82] // RealDigits[#, 10, 87] &//First (* Stefano Spezia, Jun 27 2023 *)

sumpos(n=1, asin(1/n) - sin(1/n)) \\ Michel Marcus, Jun 27 2023

Equals Sum_{k>=1} (binomial(2*k,k)/((2*k+1)*2^(2*k)) - (-1)^k/(2*k+1)!) * zeta(2*k+1). - Vaclav Kotesovec, Jun 27 2023

More terms from Stefano Spezia, Jun 27 2023

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

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A362662 Decimal expansion of Sum_{n>=1} (tan(1/n) - sin(1/n)).

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A351738 Decimal expansion of Sum_{k>0} sin(sqrt(k)) / k.

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A363906 Decimal expansion of Sum_{n>=1} (arcsin(1/n) - sin(1/n)).

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