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Fungrim entry: 014c4e

γ(SITI2log ⁣(n))<24e8n   where (S,I,T)=(k=05nHkn2k(k!)2,k=05nn2k(k!)2,14nk=02n1((2k)!)3(k!)482k(2n)2k)\left|\gamma - \left(\frac{S}{I} - \frac{T}{{I}^{2}} - \log\!\left(n\right)\right)\right| < 24 {e}^{-8 n}\; \text{ where } \left(S, I, T\right) = \left(\sum_{k=0}^{5 n} \frac{H_{k} {n}^{2 k}}{{\left(k !\right)}^{2}}, \sum_{k=0}^{5 n} \frac{{n}^{2 k}}{{\left(k !\right)}^{2}}, \frac{1}{4 n} \sum_{k=0}^{2 n - 1} \frac{{\left(\left(2 k\right)!\right)}^{3}}{{\left(k !\right)}^{4} {8}^{2 k} {\left(2 n\right)}^{2 k}}\right)
Assumptions:nZ1n \in \mathbb{Z}_{\ge 1}
References:
  • R. Brent and F. Johansson. A bound for the error term in the Brent-McMillan algorithm. Mathematics of Computation 2015, 84(295). DOI: 10.1090/S0025-5718-2015-02931-7
TeX:
\left|\gamma - \left(\frac{S}{I} - \frac{T}{{I}^{2}} - \log\!\left(n\right)\right)\right| < 24 {e}^{-8 n}\; \text{ where } \left(S, I, T\right) = \left(\sum_{k=0}^{5 n} \frac{H_{k} {n}^{2 k}}{{\left(k !\right)}^{2}}, \sum_{k=0}^{5 n} \frac{{n}^{2 k}}{{\left(k !\right)}^{2}}, \frac{1}{4 n} \sum_{k=0}^{2 n - 1} \frac{{\left(\left(2 k\right)!\right)}^{3}}{{\left(k !\right)}^{4} {8}^{2 k} {\left(2 n\right)}^{2 k}}\right)

n \in \mathbb{Z}_{\ge 1}
Definitions:
Fungrim symbol Notation Short description
Absz\left|z\right| Absolute value
ConstGammaγ\gamma The constant gamma (0.577...)
Powab{a}^{b} Power
Loglog ⁣(z)\log\!\left(z\right) Natural logarithm
Expez{e}^{z} Exponential function
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
Factorialn!n ! Factorial
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("014c4e"),
    Formula(Where(Less(Abs(Sub(ConstGamma, Sub(Sub(Div(S, I), Div(T, Pow(I, 2))), Log(n)))), Mul(24, Exp(Neg(Mul(8, n))))), Equal(Tuple(S, I, T), Tuple(Sum(Div(Mul(HarmonicNumber(k), Pow(n, Mul(2, k))), Pow(Factorial(k), 2)), Tuple(k, 0, Mul(5, n))), Sum(Div(Pow(n, Mul(2, k)), Pow(Factorial(k), 2)), Tuple(k, 0, Mul(5, n))), Mul(Div(1, Mul(4, n)), Sum(Div(Pow(Factorial(Mul(2, k)), 3), Mul(Mul(Pow(Factorial(k), 4), Pow(8, Mul(2, k))), Pow(Mul(2, n), Mul(2, k)))), Tuple(k, 0, Sub(Mul(2, n), 1)))))))),
    Variables(n),
    Assumptions(Element(n, ZZGreaterEqual(1))),
    References("R. Brent and F. Johansson. A bound for the error term in the Brent-McMillan algorithm. Mathematics of Computation 2015, 84(295). DOI: 10.1090/S0025-5718-2015-02931-7"))

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Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-08-17 11:32:46.829430 UTC