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 1-4 of 4 results.

A309209 Continued fraction expansion of the negation of the constant defined by A265011 (among other formulas, this is Sum_{k >= 1} arctan((-1)^k/k)).

Original entry on oeis.org

-1, 2, 36, 1, 40, 1, 94, 1, 1
Offset: 0

Views

Author

N. J. A. Sloane, Jul 29 2019; definition corrected Jul 30 2019 (Thanks to Jianing Song for pointing out that something was wrong.)

Keywords

Links

Crossrefs

Cf. A265011.

Extensions

Name corrected by Jianing Song, Aug 30 2019

A343469 Decimal expansion of Sum_{n>=1} (-1)^(n-1)/(n*arctan(n)).

Original entry on oeis.org

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

Views

Author

Bernard Schott, Apr 16 2021

Keywords

Comments

The alternating series test shows this series is convergent.

Examples

			0.98085427459135061539223227829901962211854180072023203606979...

Links

Crossrefs

Cf. A343470, A265011.

Programs

  • Maple
    evalf(sum((-1)^(n-1)/(n*arctan(n)), n=1..infinity),120);
  • PARI
    sumalt(n=1, (-1)^(n-1)/(n*atan(n))) \\ Michel Marcus, Apr 16 2021

Formula

Equals Sum_{n>=1} (-1)^(n-1)/(n*arctan(n)).

A343470 Decimal expansion of Sum_{n>=1} ((-1)^(n-1))*arctan(n)/n.

Original entry on oeis.org

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

Views

Author

Bernard Schott, Apr 17 2021

Keywords

Comments

The alternating series test shows the series is convergent.

Examples

			0.46571230352630363552682770124023013687237216061516248...

Links

Crossrefs

Cf. A343469, A265011.

Programs

  • Maple
    evalf(sum(((-1)^(n-1))*arctan(n)/n, n=1..infinity),120);
  • PARI
    sumalt(n=1, (-1)^(n-1)*atan(n)/n) \\ Michel Marcus, Apr 17 2021

Formula

Equals Sum_{n>=1} ((-1)^(n-1))*arctan(n)/n.

A355921 Decimal expansion of Sum_{k>=1} (1/k)*arctan(1/k).

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Jul 21 2022

Keywords

Examples

			1.40586929828778091125539861...

Links

Crossrefs

Cf. A091007, A265011, A343469, A343470, A352619, A355922.

Programs

  • Mathematica
    RealDigits[N[Sum[ArcTan[1/k]/k, {k, 1, Infinity}], 30], 10, 27][[1]]
  • PARI
    default(realprecision, 200); sumalt(k=1,(-1)^(k+1)*zeta(2*k)/(2*k-1)) \\ Vaclav Kotesovec, Jul 21 2022

Formula

Equals Sum_{k>=1} arccot(k)/k.
Equals Sum_{k>=1} (-1)^(k+1)*zeta(2*k)/(2*k-1).
Equals (1/2) * Integral_{x=0..1} (coth(Pi*x)*Pi/x - 1/x^2) dx.
Equals Integral_{x>=0} Si(x)/(exp(x)-1) dx, where Si(x) is the sine integral function.
Equals -Integral_{x>=0} sin(x)*log(1-exp(-x))/x dx.

Extensions

More terms from Jinyuan Wang, Jul 21 2022
Showing 1-4 of 4 results.

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

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