A397228 - OEIS

A397228

Decimal expansion of Sum_{k>=1} (-1)^(k+1) * AH(k)/k^8, where AH(k) = A058313(k)/A058312(k) is the k-th alternating harmonic (or skew-harmonic) number.

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

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OFFSET

0,1

LINKS

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

Ali Shadhar Olaikhan, An Introduction to the Harmonic Series and Logarithmic Integrals, 2nd ed., 2023, section 4.1, p. 261.

Cornel Ioan Vălean, More (Almost) Impossible Integrals, Sums, and Series, Springer Cham, 2023. See section 4.22, p. 427.

Index entries for sequences related to harmonic numbers.

FORMULA

Equals 255*log(2)*zeta(8)/128 + 63*zeta(2)*zeta(7)/64 + 3*zeta(3)*zeta(6)/4 + 15*zeta(4)*zeta(5)/16 - 2039*zeta(9)/512.

EXAMPLE

0.998166732747089980354491711435044445451876830266037...

MATHEMATICA

RealDigits[255*Log[2]*Zeta[8]/128 + 63*Zeta[2]*Zeta[7]/64 + 3*Zeta[3]*Zeta[6]/4 + 15*Zeta[4]*Zeta[5]/16 - 2039*Zeta[9]/512, 10, 120][[1]]

PROG

(PARI) 255*log(2)*zeta(8)/128 + 63*zeta(2)*zeta(7)/64 + 3*zeta(3)*zeta(6)/4 + 15*zeta(4)*zeta(5)/16 - 2039*zeta(9)/512

CROSSREFS

Cf. A058312, A058313.

Cf. A002117, A002162, A013661, A013662, A013663, A013664, A013665, A013666.

Sum_{k>=1} (-1)^(k+1) * AH(k)/k^m: A369884 (m=1), A397225 (m=2), A397226 (m=4), A397227 (m=6), this constant (m=8).

Sequence in context: A292825 A347058 A091667 * A300284 A347057 A334446

Adjacent sequences: A397225 A397226 A397227 * A397229 A397230 A397231

KEYWORD

nonn,cons

AUTHOR

Amiram Eldar, Jun 19 2026

STATUS

approved