A391784 Decimal expansion of Sum_{k>=1} log(1 + 1/(k+1)) / k.

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

Offset

0,1

Comments

Denoted by M_1 in Boyadzhiev (2018).

Links

Table of n, a(n) for n=0..104.

Khristo N. Boyadzhiev, A special constant and series with zeta values and harmonic numbers, Gazeta Matematica, Seria A, Vol. 115, No. 3-4 (2018), pp. 1-16; arXiv preprint, arXiv:1903.11141 [math.NT], 2019. See Proposition 8, p. 9.

Khristo N. Boyadzhiev, New Series Identities with Cauchy, Stirling, and Harmonic Numbers, and Laguerre Polynomials, Journal of Integer Sequences, Vol. 23 (2020), Article 20.11.7. See p. 9.

Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2024, p. 19.

Formula

Formulas from Boyadzhiev (2018):

Equals Integral_{x=1..2} (psi(1+x) + gamma)/x dx, where psi is the digamma function and gamma is Euler's constant (A001620).

Equals Sum_{k>=1} H^{-}(k) * (zeta(k+1) - 1), where H^{-}(k) = A058313(k)/A058312(k) is the k-th alternating harmonic (or skew-harmonic) number.

Equals Sum_{k>=1} ((-1)^k/k) * (k - zeta(2) - zeta(3) - ... - zeta(k)).

Formula from Boyadzhiev (2020):

Equals 1/2 + Sum_{k>=0} (-1)^(k+1) * c(k) * H(k) / (k+1)!, where c(k) = A006232(k)/A006233(k) is the k-th Cauchy number, and H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

Example

0.86062019285313836404349492027458782013639448695552...

Mathematica

seq[dig_] := RealDigits[NIntegrate[(PolyGamma[1 + x] + EulerGamma)/x, {x, 1, 2}, WorkingPrecision -> dig], 10, dig][[1]]; seq[120]

Prog

(PARI) sumpos(k = 1, log(1 + 1/(k+1)) / k)

Crossrefs

Cf. A001620, A001008, A002805, A006232, A006233, A058313, A058312, A391783.

Sequence in context: A153617 A069855 A156551 * A074738 A344041 A240805

Adjacent sequences: A391781 A391782 A391783 * A391785 A391786 A391787

Keyword

nonn,cons

Author

Amiram Eldar, Dec 20 2025

Status

approved