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Fungrim entry: 3c4480

ezk=0N1zkk!zNN!(1zN)\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:zCandNZandN>zz \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
Absz\left|z\right| Absolute value
Expez{e}^{z} Exponential function
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
Powab{a}^{b} Power
Factorialn!n ! Factorial
CCC\mathbb{C} Complex numbers
ZZZ\mathbb{Z} Integers
Source code for this entry:
Entry(ID("3c4480"),
    Formula(LessEqual(Abs(Sub(Exp(z), Sum(Div(Pow(z, k), Factorial(k)), Tuple(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)))))

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

2019-08-21 11:44:15.926409 UTC