A096418 -id:A096418 - 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

A342680 Decimal expansion of Sum_{n>=1} sin(sin(n)/n).

Original entry on oeis.org

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

Offset: 0

Bernard Schott, Mar 18 2021

Abel summation shows the series is convergent.

			0.96139431594573654724764595316154730686858269301058...

Konrad Knopp, Theory and Application of Infinite Series, Blackie, 1928, p. 313.
Jean-Marie Monier, Analyse, Tome 3, 2ème année, MP.PSI.PC.PT, Dunod, 1997, Exercice C.3.7 2.3.b)4. p. 309.

Wikipedia, Summation by parts.

Cf. A070751, A070752, A096418, A248946.

nDgtsOutput:=110; nDgtsPrecision:=nDgtsOutput+10; SetDefaultRealField(RealField(nDgtsPrecision)); kMax:=Ceiling(1.395*nDgtsPrecision-3); mMax:=Ceiling(1.5*kMax); sum:=0.0; S1:=[0.0 : j in [1..kMax]]; n:=0; for m in [1..mMax] do S2:=S1; for k in [1..355] do n:=n+1; sum+:=Sin(Sin(n)/n); end for; S1[1]:=sum; for j in [1..kMax-1] do S1[j+1]:=(S2[j]+S1[j])/2; end for; end for; ChangePrecision(S1[#S1], nDgtsOutput); // The constants 1.395 and 1.5 were empirically derived; 355 is used because 355/Pi is very close to an odd integer. - Jon E. Schoenfield, Mar 21 2021

a(3)-a(104) from Jon E. Schoenfield, Mar 20 2021

A122143 Decimal expansion of Sum_{k >= 1} cos(k)/k^2.

Original entry on oeis.org

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

Offset: 0

T. D. Noe, Aug 28 2006

Also, decimal expansion of the real part of Sum_{k>=1} e^(i*k)/k^2. [Bruno Berselli, Mar 24 2013]

			0.324137740053329817241093475006273747120365201519245527248085933216...

Cf. A096418 (decimal expansion of Sum_{k >= 1} sin(k)/k^2).

Cf. A263192, A121225.

Print[x=FullSimplify[Sum[Cos[n]/n^2, {n,Infinity}]]]; RealDigits[N[x,110]][[1]]

(2*Pi*(Pi-3)+3)/12 \\ Jianing Song, Nov 09 2019

Equals (2*Pi*(Pi-3)+3)/12.

A223709 Decimal expansion of (Pi-1)(2Pi-1)/12.

Original entry on oeis.org

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

Offset: 0

Bruno Berselli, Mar 26 2013

Let p = sum(sin(k)/k, k>=1) = (Pi-1)/2 (A096444) and q = sum(sin(k/2)/k, k>=1) = (2*Pi-1)/4, then A223709 = (2/3)*p*q.

This is the case h=1 in sum(sin(k/h)/k^3, k>=1) = (h*Pi-1)*(2h*Pi-1)/(12*h^3) = ((h*Pi-1)/(2h))*((2h*Pi-1)/(4h))*(2/(3h)), where (j*Pi-1)/(2j) = sum(sin(k/j)/k, k>=1) and 1/j is real but not an integer multiple of 2Pi.

			0.9428692367841114601900876541594828015029908846963553...

Tom M. Apostol, Calculus, Vol. 1, John Wiley & Sons, 1967 (2nd ed.). This constant is the case s=1, t=3 in sum(sin(n*s)/n^t, n>=1), see p. 409.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for transcendental numbers

Cf. A096418, A096444, A217909, A223710.

RealDigits[(Pi - 1) (2 Pi - 1)/12, 10, 90][[1]]

Equals sum(sin(k)/k^3, k>=1).

A329247 Decimal expansion of Sum_{k>=1} cos(k*Pi/6)/k.

Original entry on oeis.org

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

Offset: 0

Jianing Song, Nov 09 2019

Sum_{k>=1} cos(k*x)/k = Re(Sum_{k>=1} exp(k*x*i)/k) = Re(-log(1-exp(x*i))) = -log(2*|sin(x/2)|), x != 2*m*Pi, where i is the imaginary unit.
In general, for real s and complex z, let f(s,z) = Sum_{k>=1} z^k/k^s, then:
(a) if s <= 0, then f(s,z) converges to Polylog(s,z) if |z| < 1;
(b) if 0 < s <= 1, then f(s,z) converges to Polylog(s,z) if z != 1;
(c) if s > 1, then f(s,z) converges to Polylog(s,z) if |z| <= 1.
As a result, let z = e^(i*x), then the series Sum_{k>=1} (cos(k*x) + i*sin(k*x))/k^s converges to Polylog(s,e^(i*x)) if and only if s > 1, or 0 < s <= 1 and x != 2*m*Pi.

			0.65847894846240835431252317365398422201349098573375...

Cornel Ioan Vălean, Problem 11930, The American Mathematical Monthly, Vol. 123, No. 8 (2016), p. 831; A Telescoping Series with Inverse Hyperbolic Sine, Solution to Problem 11930 by Ángel Plaza, ibid., Vol. 125, No. 6 (2018), pp. 568-569.

Similar sequences:
A263192 (Sum_{k>=1} cos(k)/sqrt(k) = Re(Polylog(1/2,exp(i))));
A263193 (Sum_{k>=1} sin(k)/sqrt(k) = Im(Polylog(1/2,exp(i))));
this sequence (Sum_{k>=1} cos(k*Pi/6)/k = Re(Polylog(1,exp(i*Pi/6))));
A121225 (Sum_{k>=1} cos(k)/k = Re(Polylog(1,exp(i))));
A329246 (Sum_{k>=1} cos(k*Pi/4)/k = Re(Polylog(1,exp(i*Pi/4))));
A096444 (Sum_{k>=1} sin(k)/k = Im(Polylog(1,exp(i))));
A122143 (Sum_{k>=1} cos(k)/k^2 = Re(Polylog(2,exp(i))));
A096418 (Sum_{k>=1} sin(k)/k^2 = Im(Polylog(2,exp(i)))).

Digits := 100: (log(2 + sqrt(3))/2)*10^91:
ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Nov 09 2019

RealDigits[Log[2 + Sqrt[3]]/2, 10, 100][[1]] (* Amiram Eldar, Dec 05 2021 *)

default(realprecision, 100); log(2 + sqrt(3))/2

Equals log(2 + sqrt(3))/2.
Equals -log(2*sin(Pi/12)).
Equals arccoth(sqrt(3)). - Amiram Eldar, Dec 05 2021
From Amiram Eldar, Mar 26 2022: (Start)
Equals arcsinh(1/sqrt(2)).
Equals Sum_{n>=1} arcsinh(1/(sqrt(2^(n+2)+2)+sqrt(2^(n+1)+2))) (Vălean, 2106). (End)
log(2 + sqrt(3))/2 = Sum_{n >= 1} 1/(n*P(n, sqrt(3))*P(n-1, sqrt(3))), where P(n, x) denotes the n-th Legendre polynomial. The first ten terms of the series gives the approximation log(2 + sqrt(3))/2  = 0.658478948(35...) correct to 9 decimal places. - Peter Bala, Mar 16 2024

A329246 Decimal expansion of Sum_{k>=1} cos(k*Pi/4)/k.

Original entry on oeis.org

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

Offset: 0

Jianing Song, Nov 09 2019

Sum_{k>=1} cos(k*x)/k = Re(Sum_{k>=1} exp(k*x*i)/k) = Re(-log(1-exp(x*i))) = -log(2*|sin(x/2)|), x != 2*m*Pi, where i is the imaginary unit.
In general, for real s and complex z, let f(s,z) = Sum_{k>=1} z^k/k^s, then:
(a) if s <= 0, then f(s,z) converges to Polylog(s,z) if |z| < 1;
(b) if 0 < s <= 1, then f(s,z) converges to Polylog(s,z) if z != 1;
(c) if s > 1, then f(s,z) converges to Polylog(s,z) if |z| <= 1.
As a result, let z = e^(i*x), then the series Sum_{k>=1} (cos(k*x) + i*sin(k*x))/k^s converges to Polylog(s,e^(i*x)) if and only if s > 1, or 0 < s <= 1 and x != 2*m*Pi.

			Sum_{k>=1} cos(k*Pi/4)/k = -log(2*|sin(Pi/8)|) = 0.2673999983...

Similar sequences:
A263192 (Sum_{k>=1} cos(k)/sqrt(k) = Re(Polylog(1/2,exp(i))));
A263193 (Sum_{k>=1} sin(k)/sqrt(k) = Im(Polylog(1/2,exp(i))));
A329247 (Sum_{k>=1} cos(k*Pi/6)/k = Re(Polylog(1,exp(i*Pi/6))));
A121225 (Sum_{k>=1} cos(k)/k = Re(Polylog(1,exp(i))));
this sequence (Sum_{k>=1} cos(k*Pi/4)/k = Re(Polylog(1,exp(i*Pi/4))));
A096444 (Sum_{k>=1} sin(k)/k = Im(Polylog(1,exp(i))));
A122143 (Sum_{k>=1} cos(k)/k^2 = Re(Polylog(2,exp(i))));
A096418 (Sum_{k>=1} sin(k)/k^2 = Im(Polylog(2,exp(i)))).

RealDigits[Log[1 + Sqrt[2]/2]/2, 10, 120][[1]] (* Amiram Eldar, May 31 2023 *)

default(realprecision, 100); log(1 + sqrt(2)/2)/2

Equals log(1 + sqrt(2)/2)/2.

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

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

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A342680 Decimal expansion of Sum_{n>=1} sin(sin(n)/n).

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A122143 Decimal expansion of Sum_{k >= 1} cos(k)/k^2.

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A223709 Decimal expansion of (Pi-1)*(2*Pi-1)/12.

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A329247 Decimal expansion of Sum_{k>=1} cos(k*Pi/6)/k.

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A329246 Decimal expansion of Sum_{k>=1} cos(k*Pi/4)/k.

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

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A223709 Decimal expansion of (Pi-1)(2Pi-1)/12.