A113024 Decimal expansion of Sum_{k>=1} -(-1)^k/sqrt(k).

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

Offset: 0

Author

Robert G. Wilson v, Oct 11 2005

Keywords

• cons
• nonn

Examples

			1 - 1/sqrt(2) + 1/sqrt(3) - 1/sqrt(4) + 1/sqrt(5) - 1/sqrt(6) + 1/sqrt(7) ... =
0.60489864342163037024726591423595549975976254513024738037854664808...
		

References

• Stephen Fletcher Hewson, A Mathematical Bridge: An Intuitive Journey In Higher Mathematics, World Scientific, NJ, 2003, p. 83.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• Rick Kreminski, Using Simpson's rule to approximate sums of infinite series, Coll. Math. J. 28 (5) (1997), p 368-376, Table 1.
• Eric Weisstein's World of Mathematics, Zeta Function.

Crossrefs

Cf. A002193, A059750, A263192, A263193, A265162.

Programs

Maple

Zeta(0,1/2,1/2); evalf(%) ; # R. J. Mathar, Dec 17 2024

Mathematica

RealDigits[(1 - Sqrt[2])Zeta[1/2], 10, 111][[1]]

PARI

(1-sqrt(2))*zeta(1/2) \\ G. C. Greubel, Apr 09 2018

Formula

Equals (1-sqrt(2))*zeta(1/2) = (-1+A002193) * A059750.

A265162/A113024 = gamma/2 + Pi/4 - (1/2 + sqrt(2))*log(2) + log(Pi)/2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Dec 03 2015

Equals -zeta(1/2, 1/2). - Peter Luschny, Nov 03 2020