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Fungrim entry: d3baaf

(nk)<12πnk(nk)nnkk(nk)nk{n \choose k} < \frac{1}{\sqrt{2 \pi}} \sqrt{\frac{n}{k \left(n - k\right)}} \frac{{n}^{n}}{{k}^{k} {\left(n - k\right)}^{n - k}}
Assumptions:nZ2andk{1,2,,n1}n \in \mathbb{Z}_{\ge 2} \,\mathbin{\operatorname{and}}\, k \in \{1, 2, \ldots, n - 1\}
TeX:
{n \choose k} < \frac{1}{\sqrt{2 \pi}} \sqrt{\frac{n}{k \left(n - k\right)}} \frac{{n}^{n}}{{k}^{k} {\left(n - k\right)}^{n - k}}

n \in \mathbb{Z}_{\ge 2} \,\mathbin{\operatorname{and}}\, k \in \{1, 2, \ldots, n - 1\}
Definitions:
Fungrim symbol Notation Short description
Binomial(nk){n \choose k} Binomial coefficient
Sqrtz\sqrt{z} Principal square root
ConstPiπ\pi The constant pi (3.14...)
Powab{a}^{b} Power
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
Range{a,a+1,,b}\{a, a + 1, \ldots, b\} Integers between given endpoints
Source code for this entry:
Entry(ID("d3baaf"),
    Formula(Less(Binomial(n, k), Mul(Mul(Div(1, Sqrt(Mul(2, ConstPi))), Sqrt(Div(n, Mul(k, Sub(n, k))))), Div(Pow(n, n), Mul(Pow(k, k), Pow(Sub(n, k), Sub(n, k))))))),
    Variables(n, k),
    Assumptions(And(Element(n, ZZGreaterEqual(2)), Element(k, Range(1, Sub(n, 1))))))

Topics using this entry

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

2019-10-05 13:11:19.856591 UTC