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 A224196 Decimal expansion of the 3rd du Bois-Reymond constant. 11
 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, 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, 3, 6, 3, 7, 2, 3, 3, 3, 0, 5, 7, 5, 1, 4, 9, 3, 7, 4, 7 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS From Jon E. Schoenfield, Aug 17 2014: (Start) Evaluating the partial sums Sum_{k=1..j} 2*(1 + x_k ^ 2)^(-3/2) (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 c0 + c2/j^2 + c3/j^3 + c4/j^4 + ... where c0 = 0.02825176416006793787321073299629898515427... and c2 and c3 are -4/Pi^3 and 16/Pi^3, respectively. (The k-th root of tan(t)=t is r - d_1/r - d_2/r^3 - d_3/r^5 - d_4/r^7 - d_5/r^9 - ... where r = (k+1/2) * Pi and d_j = A079330(j)/A088989(j).) (End) 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 REFERENCES S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 237-239. LINKS Jon E. Schoenfield and Vaclav Kotesovec, Table of n, a(n) for n = 0..450 (first 100 terms from Jon E. Schoenfield) Eric Weisstein's World of Mathematics, du Bois-Reymond Constants. EXAMPLE 0.028251764... MATHEMATICA 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 CROSSREFS Cf. A062546 (2nd), A207528 (4th), A243108 (5th), A245333 (6th). Sequence in context: A021358 A332353 A203022 * A141449 A065485 A229939 Adjacent sequences: A224193 A224194 A224195 * A224197 A224198 A224199 KEYWORD nonn,cons AUTHOR Jean-François Alcover, Apr 15 2013 EXTENSIONS a(8)-a(15) from Robert G. Wilson v, Nov 06 2013 More terms from Jon E. Schoenfield, Aug 17 2014 STATUS approved

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Last modified May 28 01:33 EDT 2023. Contains 362992 sequences. (Running on oeis4.)