|
%I #27 May 03 2023 15:16:34
%S 5,9,1,5,6,0,7,7,9,3,4,9,8,1,7,3,4,0,2,1,3,8,4,6,9,0,3,3,4,5,3,4,3,4,
%T 6,9,5,6,2,3,5,3,8,9,6,2,5,4,5,6,7,1,7,4,6,8,1,0,7,6,8,4,5,9,1,6,5,5,
%U 7,9,8,0,5,3,0,2,4,9,5,9,0,8,3,6,2,7,0,4,7,2,9,0,7,8,7,6,2,7,6,9,7,8,3,8,2,7
%N Decimal expansion of Sum_{k>=1} (-1)^(k+1)*floor(sqrt(k))/k.
%C If the floor function is replaced by the fractional part function, then Sum_{k>=1} (-1)^(k+1)*frac(sqrt(k))/k = (A113024 - (this constant)) = 0.01333786407...
%H Wolfgang Hintze and River Li, <a href="https://math.stackexchange.com/questions/3450405/closed-expression-for-sum-sum-k-1-infty-1k1-frac-left-lfloor-sqr">Closed expression for sum Sum_{k=1..oo} (-1)^(k+1)*floor(sqrt(k))/k</a>, Mathematics Stackexchange, 2019.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>.
%F Equals log(2) + Sum_{n>=1} (-1)^n*n*Sum_{i=1..n} 1/((n^2+2*i-1)*(n^2+2*i)) (Li, 2019).
%F Equals Integral_{x=0..1} (1-theta_4(0,x))/(2*x*(x+1)), where theta_4(z, q) is the 4th Jacobi theta function (Hintze, 2019).
%e 0.591560779349817340213846903345...
%p evalf(log(2) + Sum((-1)^n*n*Sum(1/((n^2 + 2*i - 1)*(n^2 + 2*i)), i = 1..n), n = 1..infinity), 200); # _Vaclav Kotesovec_, May 02 2023
%t RealDigits[NIntegrate[(1 - EllipticTheta[4, x])/(2*x*(x + 1)), {x, 0, 1}, WorkingPrecision -> 30]][[1]]
%o (PARI) default(realprecision, 200); log(2) + sumalt(n=1, (-1)^n*n*sum(i=1, n, 1/((n^2 + 2*i - 1)*(n^2 + 2*i)) )) \\ _Vaclav Kotesovec_, May 02 2023
%Y Cf. A000196, A113024.
%K nonn,cons
%O 0,1
%A _Amiram Eldar_, May 02 2023
%E More digits from _Vaclav Kotesovec_, May 02 2023
|
