A224196 Decimal expansion of the 3rd du Bois-Reymond constant.

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

Author

Jean-François Alcover, Apr 15 2013

Keywords

• nonn
• cons

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

Examples

			0.028251764...
		

References

• Steven 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.

Crossrefs

Cf. A062546 (2nd), A207528 (4th), A243108 (5th), A245333 (6th).

Programs

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

Extensions

a(8)-a(15) from Robert G. Wilson v, Nov 06 2013

More terms from Jon E. Schoenfield, Aug 17 2014