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%I #84 Feb 24 2023 04:42:55
%S 0,2,8,2,5,1,7,6,4,1,6,0,0,6,7,9,3,7,8,7,3,2,1,0,7,3,2,9,9,6,2,9,8,9,
%T 8,5,1,5,4,2,7,0,2,0,2,0,1,8,1,6,0,9,9,1,7,7,1,6,9,1,9,4,8,2,9,4,4,6,
%U 3,6,3,7,2,3,3,3,0,5,7,5,1,4,9,3,7,4,7
%N Decimal expansion of the 3rd du Bois-Reymond constant.
%C From _Jon E. Schoenfield_, Aug 17 2014: (Start)
%C Evaluating the partial sums
%C Sum_{k=1..j} 2*(1 + x_k ^ 2)^(-3/2)
%C (where x_k is the k-th root of tan(t)=t; see the Mathworld link) at j = 1, 2, 4, 8, 16, 32, ..., it becomes apparent that they approach
%C c0 + c2/j^2 + c3/j^3 + c4/j^4 + ...
%C where
%C c0 = 0.02825176416006793787321073299629898515427...
%C and c2 and c3 are -4/Pi^3 and 16/Pi^3, respectively.
%C (The k-th root of tan(t)=t is
%C r - d_1/r - d_2/r^3 - d_3/r^5 - d_4/r^7 - d_5/r^9 - ...
%C where r = (k+1/2) * Pi and d_j = A079330(j)/A088989(j).) (End)
%C d_n = A079330(n)/A088989(n) ~ Gamma(1/3) / (2^(2/3) * 3^(1/6) * Pi^(5/3)) * (Pi/2)^(2*n) / n^(4/3). - _Vaclav Kotesovec_, Aug 19 2014
%D S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 237-239.
%H Jon E. Schoenfield and Vaclav Kotesovec, <a href="/A224196/b224196.txt">Table of n, a(n) for n = 0..450</a> (first 100 terms from Jon E. Schoenfield)
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/duBois-ReymondConstants.html">du Bois-Reymond Constants</a>.
%e 0.028251764...
%t digits = 16; m0 = 10^5; dm = 10^5; Clear[xi, c3]; xi[n_?NumericQ] := xi[n] = x /. FindRoot[x == Tan[x], {x, n*Pi + Pi/2 - 1/(4*n)}, WorkingPrecision -> digits + 5]; c3[m_] := c3[m] = 2*Sum[1/(1 + xi[n]^2)^(3/2), {n, 1, m}] - 2*PolyGamma[2, m + 1]/(2*Pi^3); c3[m0] ; c3[m = m0 + dm]; While[RealDigits[c3[m], 10, digits] != RealDigits[c3[m - dm], 10, digits], Print["m = ", m, " ", c3[m]]; m = m + dm]; RealDigits[c3[m], 10, digits] // First
%Y Cf. A062546 (2nd), A207528 (4th), A243108 (5th), A245333 (6th).
%K nonn,cons
%O 0,2
%A _Jean-François Alcover_, Apr 15 2013
%E a(8)-a(15) from _Robert G. Wilson v_, Nov 06 2013
%E More terms from _Jon E. Schoenfield_, Aug 17 2014
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