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Decimal expansion of Sum_{n>=1} (-1)^(n-1)*H(n)/n^2, where H(n) is the n-th harmonic number.
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%I #24 Nov 03 2024 09:33:24

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

%T 4,4,2,2,8,1,1,6,4,3,2,7,1,2,8,1,1,8,0,1,1,2,0,1,6,9,7,2,2,0,8,8,6,4,

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

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

%D Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 43.

%H R. Barbieri, J. A. Mignaco, and E. Remiddi, <a href="https://dx.doi.org/10.1007/BF02728545">Electron form factors up to fourth order. I.</a>, Il Nuovo Cim. 11A (4) (1972) 824-864, table II (13)

%H Philippe Flajolet and Bruno Salvy, <a href="http://algo.inria.fr/flajolet/Publications/FlSa98.pdf">Euler Sums and Contour Integral Representations</a>, Experimental Mathematics 7:1 (1998), page 32.

%H Paul J. Nahin, <a href="https://doi.org/10.1007/978-3-030-43788-6">Inside interesting integrals</a>, Undergrad. Lecture Notes in Physics, Springer (2020), (C5.15)

%H Michael I. Shamos, <a href="http://euro.ecom.cmu.edu/people/faculty/mshamos/cat.pdf">A catalog of the real numbers</a>, (2007). See p. 561.

%F Equals 5*zeta(3)/8.

%F Equals -Integral_{x=0..1} (log(1+x)*log(1-x)/x)*dx. - _Amiram Eldar_, May 06 2023

%F Equals Sum_{m>=1} Sum_{n>=1} (-1)^(m-1)/(m*n*(m + n)) (see Finch). - _Stefano Spezia_, Nov 02 2024

%e 0.7512855644747464283748363509446562442281164327128118011201697220886...

%t RealDigits[ 5*Zeta[3]/8, 10, 100] // First

%Y Cf. A002117 (zeta(3)), A197070 (3*zeta(3)/4), A233091 (7*zeta(3)/8), A076788 (alternating sum with denominator n), A152648 (non-alternating sum with denominator n^2), A152649 (non-alternating sum with denominator n^3), A233033 (alternating sum with denominator n^3).

%K nonn,cons

%O 0,1

%A _Jean-François Alcover_, Dec 04 2013, after the comment by _Peter Bala_ about A233033.