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

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

1,2

Links

Table of n, a(n) for n=1..87.

Mathematics Stack Exchange, Converting the sum: Sum_{n=1..oo}(1/n) * cot^(-1)(n) to an integral, 2017.

Mathematics Stack Exchange, Summing an Arctangent Series, 2021.

Michael Ian Shamos, Shamos's Catalog of the Real Numbers, 2011, p. 428.

Eric Weisstein's World of Mathematics, Sine Integral.

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.

Example

1.40586929828778091125539861...

Mathematica

RealDigits[N[Sum[ArcTan[1/k]/k, {k, 1, Infinity}], 30], 10, 27][[1]]

Prog

(PARI) default(realprecision, 200); sumalt(k=1, (-1)^(k+1)*zeta(2*k)/(2*k-1)) \\ Vaclav Kotesovec, Jul 21 2022

Crossrefs

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

Sequence in context: A198747 A021965 A372274 * A164108 A064520 A267313

Adjacent sequences: A355918 A355919 A355920 * A355922 A355923 A355924

Keyword

nonn,cons

Author

Amiram Eldar, Jul 21 2022

Extensions

More terms from Jinyuan Wang, Jul 21 2022

Status

approved