A117231 Decimal expansion of Integral_{x=0..1} cos((log x)/x)/x dx.

3, 2, 3, 3, 6, 7, 4, 3, 1, 6, 7, 7, 7, 7, 8, 7, 6, 1, 3, 9, 9, 3, 7, 0, 0, 8, 7, 9, 5, 2, 1, 7, 0, 4, 4, 6, 6, 5, 1, 0, 4, 6, 6, 2, 5, 7, 2, 5, 4, 6, 9, 6, 6, 1, 6, 8, 1, 0, 3, 6, 4, 4, 3, 4, 3, 1, 7, 9, 0, 3, 3, 7, 2, 1, 0, 6, 7, 2, 8, 9, 4, 4, 3, 1, 9, 3, 0, 3, 7, 0, 4, 6, 4, 1, 0, 2, 4, 5, 1, 3, 8, 0, 2, 8, 0

Offset

0,1

Comments

Decimal expansion of solution to problem #1 in the Trefethen challenge.

Conjecturally, the value of the Borel sum of the divergent series Sum_{n >= 1} (-1)^(n+1)*(2*n)^(2*n-1). See Finch, p. 62. - Peter Bala, Dec 28 2022

Links

Amiram Eldar, Table of n, a(n) for n = 0..10001 (from Dirk Laurie's solution)

Folkmar Bornemann, Short remarks on the solution of the SIAM 100-digit challenge, 2002.

Folkmar Bornemann, Dirk Laurie, Stan Wagon, and Jörg Waldvogel, The SIAM 100-Digit Challenge, A Study in High-Accuracy Numerical Computing, SIAM, Philadelphia, 2004.

Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2022.

Walter Gautschi, The numerical evaluation of a challenging integral, Num. Alg. 49 (1-4) (2008) 187-194.

Dirk Laurie, A Twisted Tail, a web page for the book The SIAM 100-Digit Challenge.

Eric Weisstein's World of Mathematics, Hundred-Dollar, Hundred-Digit Challenge Problems.

Wikipedia, Hundred-dollar, Hundred-digit Challenge problems.

Example

Integral_{t = 0..1} t^(-1)*cos(t^(-1)*log(t)) dt = 0.32336743167777876139937..

Mathematica

RealDigits[ NIntegrate[ ProductLog'[u]*Cos[u], {u, 0, Infinity}, WorkingPrecision -> 220]][[1]][[1 ;; 105]] (* Jean-François Alcover, Oct 26 2012, after W. Gautschi *)

Crossrefs

Cf. A117232, A117233, A117234, A117235, A117236, A117237, A117238, A117239, A117240.

Sequence in context: A087713 A089319 A140396 * A106151 A323467 A341097

Adjacent sequences: A117228 A117229 A117230 * A117232 A117233 A117234

Keyword

nonn,cons

Author

Eric W. Weisstein, Mar 05 2006

Status

approved