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

0,2

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