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 - ...
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
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