A240264 - OEIS

A240264

Decimal expansion of Sum_{n >= 1} (-1)^(n+1)*H(n,2)/n, where H(n,2) = A007406(n)/A007407(n) is the n-th harmonic number of order 2.

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

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OFFSET

0,1

COMMENTS

Let a(p,q) = Sum_{n >= 1} (-1)^(n+1)*H(n,p)/n^q, then A076788 is a(1,1), A233090 is a(1,2) and this sequence is a(2,1).

Original name: "Decimal expansion of Sum_{n >= 1} (-1)^(n+1)*H(n,2)/n^2, where H(n,2) is the n-th harmonic number of order 2", which is now the name of A396736. - Amiram Eldar, Jun 04 2026

REFERENCES

Bruce C. Berndt, Ramanujan's Notebooks Part I, Springer-Verlag, 1985. See p. 258, eq. (i).

Ali Shadhar Olaikhan, An Introduction to the Harmonic Series and Logarithmic Integrals, 2021, p. 252, eq. (4.162).

LINKS

Table of n, a(n) for n=0..99.

Cornel Ioan Vălean, (Almost) Impossible Integrals, Sums, and Series, Springer International Publishing, 2019, section 4.51, p. 308, eq. (4.80), section 5.51, p. 326, section 6.51, pp. 498-502.

Cornel Ioan Vălean, More (Almost) Impossible Integrals, Sums, and Series, Springer Cham, 2023. See section 4.22, p. 424.

Eric Weisstein's World of Mathematics, Harmonic Number, eq. (42).

Index entries for sequences related to harmonic numbers.

FORMULA

Equals zeta(3) - Pi^2/12*log(2).

Equals Sum_{n >= 1} (1/2)^n * H(n,1)/n^2, where H(n,1) = Sum_{k = 1..n} 1/k. See Berndt, p. 258. - Peter Bala, Oct 28 2021

EXAMPLE

0.631966197838167906662448232015275318156671371658172...

MATHEMATICA

Zeta[3] - Pi^2/12*Log[2] // RealDigits[#, 10, 100]& // First

PROG

(PARI) zeta(3)-log(2)*Pi^2/12 \\ Charles R Greathouse IV, Apr 03 2014