A391783 Decimal expansion of Sum_{k>=1} (-1)^(k+1) * log(1 + 1/k) / k.

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

Offset

0,1

Comments

Denoted by K in Boyadzhiev and Frontczak (2021).

Links

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

Khristo Boyadzhiev and Robert Frontczak, Series Involving Euler's Eta (or Dirichlet Eta) Function, Journal of Integer Sequences, Vol. 24 (2021), Article 21.9.1.

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

Eric Weisstein's World of Mathematics, Dirichlet Eta Function.

Wikipedia, Dirichlet eta function.

Formula

Formulas from Boyadzhiev and Frontczak (2021):

Equals Sum_{k>=1} (-1)^(k+1) * eta(k+1) / k, where eta is the Dirichlet eta function.

Equals Sum_{k>=1} H(k) * (eta(k+1) - 1), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number, and eta is the Dirichlet eta function.

Equals log(2) + Sum_{k>=1} H^{-}(k) * (eta(k+1) - eta(k+2)), where H^{-}(k) = A058313(k)/A058312(k) is the k-th alternating harmonic (or skew-harmonic) number, and eta is the Dirichlet eta function.

Equals Sum_{k>=1} ((-1)^k/k) * (k - 2*log(2) - eta(2) - eta(3) - ... - eta(k)), where eta is the Dirichlet eta function.

Equals Sum_{n>=1} (1/(n*2^n)) * (Sum_{k=1..n} binomial(n, k) * (-1)^(k+1) * eta(k+1)), where eta is the Dirichlet eta function.

Equals Integral_{x=0..1} (psi(1+x) - psi(1 + x/2))/x dx, where psi is the digamma function.

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

Equals Integral_{x>=0} Ein(x)/(exp(x)+1) dx, where Ein(x) = Sum_{k>=1} (-1)^(k+1) * x*k / (k*k!) is the modified (or complementary) exponential integral.

Example

0.55212832208549207657701214720080870894310832057623...

Mathematica

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

Prog

(PARI) sumalt(k = 1, (-1)^(k+1) * log(1 + 1/k) / k)

Crossrefs

Cf. A001008, A001620, A002805, A058312, A058313, A131688, A391784.

Sequence in context: A280868 A229160 A011501 * A319305 A196614 A319905

Adjacent sequences: A391780 A391781 A391782 * A391784 A391785 A391786

Keyword

nonn,cons

Author

Amiram Eldar, Dec 20 2025

Status

approved