Fungrim entry: 3478af

\operatorname{atan}\!\left(x\right) \le \sum_{k=0}^{2 N} \frac{{\left(-1\right)}^{k} {x}^{2 k + 1}}{2 k + 1}

Assumptions:

x \in \left[0, \infty\right) \,\mathbin{\operatorname{and}}\, N \in \mathbb{Z}_{\ge 0}

TeX:

\operatorname{atan}\!\left(x\right) \le \sum_{k=0}^{2 N} \frac{{\left(-1\right)}^{k} {x}^{2 k + 1}}{2 k + 1}

x \in \left[0, \infty\right) \,\mathbin{\operatorname{and}}\, N \in \mathbb{Z}_{\ge 0}

Definitions:

Fungrim symbol	Notation	Short description
`Atan`	$\operatorname{atan}\!\left(z\right)$	Inverse tangent
`Pow`	${a}^{b}$	Power
`ClosedOpenInterval`	$\left[a, b\right)$	Closed-open interval
`Infinity`	$\infty$	Positive infinity
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n

Source code for this entry:

Entry(ID("3478af"),
    Formula(LessEqual(Atan(x), Sum(Div(Mul(Pow(-1, k), Pow(x, Add(Mul(2, k), 1))), Add(Mul(2, k), 1)), Tuple(k, 0, Mul(2, N))))),
    Variables(x, N),
    Assumptions(And(Element(x, ClosedOpenInterval(0, Infinity)), Element(N, ZZGreaterEqual(0)))))

Topics using this entry

Inverse tangent