Fungrim entry: 3c4480

\left|{e}^{z} - \sum_{k=0}^{N - 1} \frac{{z}^{k}}{k !}\right| \le \frac{{\left|z\right|}^{N}}{N ! \left(1 - \frac{\left|z\right|}{N}\right)}

Assumptions:

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; N > \left|z\right|

TeX:

\left|{e}^{z} - \sum_{k=0}^{N - 1} \frac{{z}^{k}}{k !}\right| \le \frac{{\left|z\right|}^{N}}{N ! \left(1 - \frac{\left|z\right|}{N}\right)}

z \in \mathbb{C} \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; N > \left|z\right|

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`Exp`	${e}^{z}$	Exponential function
`Sum`	$\sum_{n} f(n)$	Sum
`Pow`	${a}^{b}$	Power
`Factorial`	$n !$	Factorial
`CC`	$\mathbb{C}$	Complex numbers
`ZZ`	$\mathbb{Z}$	Integers

Source code for this entry:

Entry(ID("3c4480"),
    Formula(LessEqual(Abs(Sub(Exp(z), Sum(Div(Pow(z, k), Factorial(k)), For(k, 0, Sub(N, 1))))), Div(Pow(Abs(z), N), Mul(Factorial(N), Sub(1, Div(Abs(z), N)))))),
    Variables(z, N),
    Assumptions(And(Element(z, CC), Element(N, ZZ), Greater(N, Abs(z)))))

Topics using this entry

Exponential function