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A362742
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Decimal expansion of Sum_{k>=1} (-1)^(k+1)*floor(sqrt(k))/k.
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0
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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
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OFFSET
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0,1
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COMMENTS
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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...
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LINKS
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FORMULA
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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).
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EXAMPLE
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0.591560779349817340213846903345...
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MAPLE
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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
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MATHEMATICA
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RealDigits[NIntegrate[(1 - EllipticTheta[4, x])/(2*x*(x + 1)), {x, 0, 1}, WorkingPrecision -> 30]][[1]]
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PROG
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(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
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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