OFFSET
1,2
COMMENTS
Here gamma = 0.5772... (A001620) is Euler's constant and Re(Psi(i)) = 0.0946... (A248177) with the psi function or the digamma function Psi(x) = d/dx log(Gamma(x)) = (1/Gamma(x))*d/dx Gamma(x) as the logarithmic derivative of the gamma function.
Value of the series 1/1 + 1/(2+3) + 1/(4+5+6) + 1/(7+8+9+10) + ... = Sum_{k>=1} 2/(k*(k^2+1)) = Sum_{k>=1} 1/A006003(k), i.e., the sum of the reciprocals of the sum of the next n natural numbers. Presumably, the question of finding the value of this series was first posed in 1889 by the Dutch mathematician Pieter Hendrik Schoute (1846-1913) and repeated in 1905.
LINKS
Prof. Schoute, Question 10184, The Educational Times, and Journal of the College of Preceptors 42 (1889), nr. 339 (July 1), p. 293; Question 10184, Ibid., 58 (1905), nr. 536 (Dec. 1), p. 534.
EXAMPLE
1.343731971048019675756781145608...
MAPLE
2*(gamma+Re(Psi(I))); evalf(%, 200);
MATHEMATICA
RealDigits[2*(EulerGamma + Re[PolyGamma[0, I]]), 10, 120][[1]] (* Amiram Eldar, Dec 20 2022 *)
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Martin Renner, Dec 20 2022
STATUS
approved