cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Aug 16 2004

Keywords

Comments

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

Examples

			1.013959132360768504294574338885914687561179280077717316877048512268137...

Links

Crossrefs

Cf. A096444, A096464.
Cf. A122143 (decimal expansion of Sum_{k >= 1} cos(k)/k^2).

Programs

  • Mathematica
    $MaxExtraPrecision = 128; RealDigits[ Chop[ N[ I/2*(PolyLog[2, E^-I] - PolyLog[2, E^I]), 105]]][[1]] (* Robert G. Wilson v, Aug 16 2004 *)
  • PARI
    imag(polylog(2,exp(I))) \\ Charles R Greathouse IV, Jul 14 2014

Extensions

More terms from Robert G. Wilson v, Aug 17 2004
Sequence checked by T. D. Noe, Aug 21 2006

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

Views

Author

Jianing Song, Nov 09 2019

Keywords

Comments

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.

Examples

			0.65847894846240835431252317365398422201349098573375...

Links

Crossrefs

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)))).

Programs

  • Maple
    Digits := 100: (log(2 + sqrt(3))/2)*10^91:
ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Nov 09 2019
  • Mathematica
    RealDigits[Log[2 + Sqrt[3]]/2, 10, 100][[1]] (* Amiram Eldar, Dec 05 2021 *)
  • PARI
    default(realprecision, 100); log(2 + sqrt(3))/2

Formula

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

Views

Author

Jianing Song, Nov 09 2019

Keywords

Comments

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.

Examples

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

Crossrefs

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)))).

Programs

  • Mathematica
    RealDigits[Log[1 + Sqrt[2]/2]/2, 10, 120][[1]] (* Amiram Eldar, May 31 2023 *)
  • PARI
    default(realprecision, 100); log(1 + sqrt(2)/2)/2

Formula

Equals log(1 + sqrt(2)/2)/2.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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