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 A102886 Decimal expansion of Serret's integral: Integral_{x=0..1} log(x+1)/(x^2+1) dx. 2
 2, 7, 2, 1, 9, 8, 2, 6, 1, 2, 8, 7, 9, 5, 0, 2, 6, 6, 3, 1, 2, 5, 8, 6, 1, 1, 2, 2, 7, 9, 7, 0, 1, 7, 4, 3, 4, 1, 7, 3, 2, 2, 9, 6, 2, 5, 4, 6, 1, 6, 0, 7, 8, 6, 7, 9, 0, 7, 2, 4, 4, 0, 6, 6, 4, 9, 2, 8, 8, 5, 6, 8, 6, 4, 7, 0, 9, 2, 7, 4, 8, 3, 0, 3, 7, 9, 1, 1, 2, 0, 2, 0, 1, 3, 3, 2, 8, 7, 8, 1, 3, 2 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Named after the French mathematician Joseph-Alfred Serret (1819-1885). - Amiram Eldar, May 30 2021 REFERENCES Eric Billault et al, MPSI- Khôlles de Maths, Ellipses, 2012, exercice 11.10, pp. 252-264. L. B. W. Jolley, Summation of Series, Dover (1961), Eq. (94) on page 18. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th edition, 4.291.8. LINKS J.-A. Serret, Note sur l'intégrale Integral_{x=0..1} log(x+1)/(x^2+1) dx, Journal de Mathématiques Pures et Appliquées, Vol. 9 (1844), page 436. Eric Weisstein's World of Mathematics, Serret's Integral. FORMULA Equals Integral_{x=0..1} arctan(x)/(x+1) dx. - Jean-François Alcover, Mar 25 2013 Equals Integral_{x=0..Pi/4} log(tan(x)+1) dx [see link J.-A. Serret and reference Billault]. - Bernard Schott, Apr 23 2020 Equals Pi*log(2)/8 = Sum_{n>0} (-1)^(n+1) * H(2n) / (2n+1) = H(2)/3 - H(4)/5 + H(6)/7 -... with H(n) = Sum_{j=1..n} 1/j the harmonic numbers. [Jolley]; improved by Bernard Schott, Apr 24 2020 Equals -Integral_{x=0..1} x*arccos(x)*log(x) dx. - Amiram Eldar, May 30 2021 EXAMPLE 0.27219826128795026631258611227970174341732296254616... MATHEMATICA RealDigits[Pi*Log[2]/8, 10, 102][[1]] (* Jean-François Alcover, May 17 2013 *) PROG (PARI) Pi*log(2)/8 \\ Michel Marcus, Apr 23 2020 (PARI) intnum(x=0, 1, log(x+1)/(x^2+1)) \\ Michel Marcus, Apr 26 2020 CROSSREFS Cf. A086054 (Pi*log(2)). Sequence in context: A095194 A254251 A095711 * A204382 A072981 A023399 Adjacent sequences:  A102883 A102884 A102885 * A102887 A102888 A102889 KEYWORD nonn,cons,easy AUTHOR Eric W. Weisstein, Jan 15 2005 STATUS approved

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Last modified November 27 16:20 EST 2021. Contains 349394 sequences. (Running on oeis4.)