A113024 - OEIS

%I #30 Dec 17 2024 08:36:39

%S 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,

%T 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,

%U 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

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

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

%H G. C. Greubel, <a href="/A113024/b113024.txt">Table of n, a(n) for n = 0..10000</a>

%H Rick Kreminski, <a href="http://www.jstor.org/stable/2687066">Using Simpson's rule to approximate sums of infinite series</a>, Coll. Math. J. 28 (5) (1997), p 368-376, Table 1.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ZetaFunction.html">Zeta Function</a>.

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

%F 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

%F Equals -zeta(1/2, 1/2). - _Peter Luschny_, Nov 03 2020

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

%e 0.60489864342163037024726591423595549975976254513024738037854664808...

%p Zeta(0,1/2,1/2); evalf(%) ; # _R. J. Mathar_, Dec 17 2024

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

%o (PARI) (1-sqrt(2))*zeta(1/2) \\ _G. C. Greubel_, Apr 09 2018

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

%K cons,nonn

%O 0,1

%A _Robert G. Wilson v_, Oct 11 2005