Fungrim entry: 1eeccf

\operatorname{atan}(x) \ge \sum_{k=0}^{2 N + 1} \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}(x) \ge \sum_{k=0}^{2 N + 1} \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`	$\operatorname{atan}(z)$	Inverse tangent
`Sum`	$\sum_{n} f(n)$	Sum
`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("1eeccf"),
    Formula(GreaterEqual(Atan(x), Sum(Div(Mul(Pow(-1, k), Pow(x, Add(Mul(2, k), 1))), Add(Mul(2, k), 1)), For(k, 0, Add(Mul(2, N), 1))))),
    Variables(x, N),
    Assumptions(And(Element(x, ClosedOpenInterval(0, Infinity)), Element(N, ZZGreaterEqual(0)))))

Topics using this entry

Inverse tangent