A362742 Decimal expansion of Sum_{k>=1} (-1)^(k+1)*floor(sqrt(k))/k.

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, 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, 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

Offset: 0

Author

Amiram Eldar, May 02 2023

Keywords

• nonn
• cons

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...

Examples

			0.591560779349817340213846903345...
		

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.

Crossrefs

Cf. A000196, A113024.

Programs

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]]

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

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).

Extensions

More digits from Vaclav Kotesovec, May 02 2023