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%I #6 Apr 08 2026 10:56:17
%S 3,5,0,8,5,2,8,4,4,9,6,5,1,3,5,3,6,0,1,6,8,5,9,3,4,4,6,7,9,7,9,5,8,2,
%T 3,8,8,7,9,4,2,2,3,9,5,1,4,4,9,0,1,3,3,8,6,2,8,5,8,1,9,9,3,9,9,6,9,7,
%U 2,3,5,5,5,0,9,0,7,1,7,8,4,7,3,2,1,9,5,4,3,7,4,4,3,4,6,0,6,7,2,7,5,7,2,5,1
%N Decimal expansion of Sum_{k>=1} Pi/(sqrt(k)*sinh(Pi*sqrt(k))).
%H H. F. Sandham, <a href="https://doi.org/10.2307/2305007">Problem 4384</a>, The American Mathematical Monthly, Vol. 57, No. 2 (1950), p. 120; <a href="https://doi.org/10.2307/2306336">Modified Harmonic Series</a>, Solution to Problem 4384 by the proposer, ibid., Vol. 58, No. 8 (1951), p. 573.
%F Equals 1 - 1/2 - 1/3 - 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 - 1/10 - ..., with a change of sign after the reciprocal of each square (Sandham, 1950).
%F Equals 1 - Sum_{k>=1} (-1)^(k+1) * (H(k+1) - H(k)), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.
%e 0.350852844965135360168593448289293938717827305807594...
%t RealDigits[Pi * NSum[1 / (Sqrt[k] * Sinh[Pi*Sqrt[k]]), {k, 1, Infinity}, WorkingPrecision -> 120]][[1]]
%o (PARI) Pi * sumpos(k = 1, 1 / (sqrt(k) * sinh(Pi*sqrt(k))))
%Y Cf. A001008, A002805.
%Y Cf. A002162, A016655, A262023, A353874.
%K nonn,cons
%O 0,1
%A _Amiram Eldar_, Apr 08 2026