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
Examples
1.40586929828778091125539861...
Links
- 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.
Programs
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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