A369885 Decimal expansion of Sum_{k>=1} log(k+1)/k^2.
1, 8, 0, 0, 7, 5, 5, 0, 5, 6, 0, 0, 5, 2, 8, 2, 9, 9, 1, 4, 9, 6, 6, 0, 6, 0, 1, 4, 2, 1, 4, 8, 4, 3, 1, 8, 1, 4, 4, 5, 6, 6, 3, 7, 8, 3, 8, 1, 8, 4, 1, 7, 9, 3, 0, 2, 7, 1, 8, 6, 6, 7, 5, 9, 1, 7, 2, 9, 9, 8, 8, 3, 1, 7, 6, 3, 8, 6, 3, 1, 1, 8, 0, 5, 1, 5, 9, 2, 9, 8, 4, 3, 7, 8, 8, 9, 2, 4, 3, 8, 1, 0, 9, 8, 9
Offset: 1
Examples
1.80075505600528299149660601421484318144566378381841...
Links
- Bernard Candelpergher and Marc-Antoine Coppo, A new class of identities involving Cauchy numbers, harmonic numbers and zeta values, The Ramanujan Journal, Vol. 27 (2012), pp. 305-328; alternative link.
- István Mező, Problem 11793, Problems and Solutions, The American Mathematical Monthly, Vol. 121, No. 7 (2014), p. 648; A Series with Zetas, Solution to Problem 11793 by FAU Problem Solving Group, ibid., Vol. 123, No. 6 (2016), p. 620.
- Michael Ian Shamos, A catalog of the real numbers (2011), p. 616.
- Roberto Tauraso, Problem 11793.
- Wikipedia, Harmonic number: Harmonic numbers for real and complex values.
Programs
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Maple
evalf(sum((-1)^(k+1)*Zeta(k)/(k-2), k = 3 .. infinity) - Zeta(1, 2), 120)
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Mathematica
RealDigits[NIntegrate[HarmonicNumber[x]/x^2, {x, 1, Infinity}, WorkingPrecision -> 120]][[1]] -
PARI
sumpos(k = 1, log(k+1)/k^2)
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PARI
sumalt(k = 3, (-1)^(k+1) * zeta(k)/(k-2)) - zeta'(2)
Formula
Equals Integral_{x>=1} H(x)/x^2 dx, where H(x) is the harmonic number for real variable x (Shamos, 2011).
Equals -zeta'(2) + Sum_{k>=3} (-1)^(k+1)*zeta(k)/(k-2) (Mező, 2014).
Equals Sum_{k>=1} lambda(k)*H(k)/(k^2*k!) + 1 + zeta(3) - gamma * zeta(2), where lambda(k) = abs(A006232(k)/A006233(k)) is the n-th non-alternating Cauchy number, H(k) = A001008(k)/A002805(k) is the k-th harmonic number, and gamma is Euler's constant (A001620) (Candelpergher and Coppo, 2012). - Amiram Eldar, Mar 18 2024