# 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

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