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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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