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Proposition: Exponential Function

The exponential series

\[\sum_{n=0}^\infty\frac{x^n}{n!}\]

is an absolutely convergent series for every real number \(x\in\mathbb R\). It defines a function \(\exp:\mathbb R\mapsto \mathbb R\), called the exponential function for all \(x\in\mathbb R\).

\[\exp(x):=\sum_{n=0}^\infty\frac{x^n}{n!},\quad\quad x\in\mathbb R.\]

Table of Contents

Proofs: 1

1. Definition: Exponential Function of General Base
2. Proposition: Estimate for the Remainder Term of Exponential Function
3. Proposition: Functional Equation of the Exponential Function
4. Proposition: Continuity of Exponential Function
5. Proposition: \(\exp(0)=1\)
6. Proposition: Derivative of the Exponential Function
7. Proposition: Integral of the Exponential Function

Mentioned in:

Chapters: 1
Corollaries: 2 3 4 5
Definitions: 6 7 8 9 10
Examples: 11
Proofs: 12 13 14 15 16 17 18 19
Propositions: 20 21 22 23 24 25 26 27 28 29 30

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