# Fungrim entry: 4c0698

$\left|\frac{1}{\pi} - \left(12 \sum_{n=0}^{N - 1} \frac{{\left(-1\right)}^{n} \left(6 n\right)! \left(13591409 + 545140134 n\right)}{\left(3 n\right)! {\left(n !\right)}^{3} \cdot {640320}^{3 n + 3 / 2}}\right)\right| < \frac{1}{{151931373056000}^{N}}$
Assumptions:$N \in \mathbb{Z}_{\ge 0}$
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
Abs$\left|z\right|$ Absolute value
Pi$\pi$ The constant pi (3.14...)
Sum$\sum_{n} f(n)$ Sum
Pow${a}^{b}$ Power
Factorial$n !$ Factorial
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ 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))))

## Topics using this entry

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

2021-03-15 19:12:00.328586 UTC