OFFSET
0,1
COMMENTS
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...
LINKS
Wolfgang Hintze and River Li, Closed expression for sum Sum_{k=1..oo} (-1)^(k+1)*floor(sqrt(k))/k, Mathematics Stackexchange, 2019.
Eric Weisstein's World of Mathematics, Jacobi Theta Functions.
FORMULA
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).
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).
EXAMPLE
0.591560779349817340213846903345...
MAPLE
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
MATHEMATICA
RealDigits[NIntegrate[(1 - EllipticTheta[4, x])/(2*x*(x + 1)), {x, 0, 1}, WorkingPrecision -> 30]][[1]]
PROG
(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
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, May 02 2023
EXTENSIONS
More digits from Vaclav Kotesovec, May 02 2023
STATUS
approved