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Fungrim entry: 4c0698

1π(12n=0N1(1)n(6n)!(13591409+545140134n)(3n)!(n!)36403203n+3/2)<1151931373056000N\left|\frac{1}{\pi} - \left(12 \sum_{n=0}^{N - 1} \frac{{\left(-1\right)}^{n} \left(6 n\right)! \left(13591409 + 545140134 n\right)}{\left(3 n\right)! {\left(n !\right)}^{3} \cdot {640320}^{3 n + 3 / 2}}\right)\right| < \frac{1}{{151931373056000}^{N}}
Assumptions:NZ0N \in \mathbb{Z}_{\ge 0}
TeX:
\left|\frac{1}{\pi} - \left(12 \sum_{n=0}^{N - 1} \frac{{\left(-1\right)}^{n} \left(6 n\right)! \left(13591409 + 545140134 n\right)}{\left(3 n\right)! {\left(n !\right)}^{3} \cdot  {640320}^{3 n + 3 / 2}}\right)\right| < \frac{1}{{151931373056000}^{N}}

N \in \mathbb{Z}_{\ge 0}
Definitions:
Fungrim symbol Notation Short description
Absz\left|z\right| Absolute value
Piπ\pi The constant pi (3.14...)
Sumnf(n)\sum_{n} f(n) Sum
Powab{a}^{b} Power
Factorialn!n ! Factorial
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Source code for this entry:
Entry(ID("4c0698"),
    Formula(Less(Abs(Sub(Div(1, Pi), Parentheses(Mul(12, Sum(Div(Mul(Mul(Pow(-1, n), Factorial(Mul(6, n))), Add(13591409, Mul(545140134, n))), Mul(Mul(Factorial(Mul(3, n)), Pow(Factorial(n), 3)), Pow(640320, Add(Mul(3, n), Div(3, 2))))), For(n, 0, Sub(N, 1))))))), Div(1, Pow(151931373056000, N)))),
    Variables(N),
    Assumptions(Element(N, ZZGreaterEqual(0))))

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2020-04-08 16:14:44.404316 UTC