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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
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
KEYWORD
nonn,cons
AUTHOR
EXTENSIONS
a(8)-a(15) from Robert G. Wilson v, Nov 06 2013
More terms from Jon E. Schoenfield, Aug 17 2014
STATUS
approved