# Real exponential function via exponential series/Introduction/Section

## Definition

For every ${\displaystyle {}x\in \mathbb {R} }$, the series

${\displaystyle \sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}}$
is called the exponential series in ${\displaystyle {}x}$.

So this is just the series

${\displaystyle 1+x+{\frac {x^{2}}{2}}+{\frac {x^{3}}{6}}+{\frac {x^{4}}{24}}+{\frac {x^{5}}{120}}+\ldots .}$

## Theorem

For every ${\displaystyle {}x\in \mathbb {R} }$, the exponential series

${\displaystyle \sum _{n=0}^{\infty }{\frac {x^{n}}{n!}}}$
is

### Proof

For ${\displaystyle {}x=0}$, the statement is clear. Else, we consider the fraction

${\displaystyle {}\vert {\frac {\frac {x^{n+1}}{(n+1)!}}{\frac {x^{n}}{n!}}}\vert =\vert {\frac {x}{n+1}}\vert ={\frac {\vert {x}\vert }{n+1}}\,.}$

This is, for ${\displaystyle {}n\geq 2\vert {x}\vert }$, smaller than ${\displaystyle {}1/2}$. By the ratio test, we get convergence.

${\displaystyle \Box }$

Due to this property, we can define the real exponential function.

## Definition

The function

${\displaystyle \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto \exp x:=\sum _{n=0}^{\infty }{\frac {x^{n}}{n!}},}$

is called the (real)

exponential function.

The following statement is called the functional equation for the exponential function.

## Theorem

For real numbers ${\displaystyle {}x,y\in \mathbb {R} }$, the equation

${\displaystyle {}\exp {\left(x+y\right)}=\exp x\cdot \exp y\,}$

holds.

### Proof

The Cauchy product of the two exponential series is

${\displaystyle \sum _{n=0}^{\infty }c_{n},}$

where

${\displaystyle {}c_{n}=\sum _{i=0}^{n}{\frac {x^{i}}{i!}}\cdot {\frac {y^{n-i}}{(n-i)!}}\,.}$

This series is due to fact absolutely convergent and the limit is the product of the two limits. Furthermore, the ${\displaystyle {}n}$-th summand of the exponential series of ${\displaystyle {}x+y}$ equals

${\displaystyle {}{\frac {(x+y)^{n}}{n!}}={\frac {1}{n!}}\sum _{i=0}^{n}{\binom {n}{i}}x^{i}y^{n-i}=c_{n}\,,}$

so that both sides coincide.

${\displaystyle \Box }$

## Corollary

${\displaystyle \mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto \exp x,}$
fulfills the following properties.
1. ${\displaystyle {}\exp 0=1}$.
2. For every ${\displaystyle {}x\in \mathbb {R} }$, we have ${\displaystyle {}\exp {\left(-x\right)}=(\exp x)^{-1}}$. In particular ${\displaystyle {}\exp x\neq 0}$.
3. For integers ${\displaystyle {}n\in \mathbb {Z} }$, the relation ${\displaystyle {}\exp n=(\exp 1)^{n}}$ holds.
4. For every ${\displaystyle {}x}$, we have ${\displaystyle {}\exp x\in \mathbb {R} _{+}}$.
5. For ${\displaystyle {}x>0}$ we have ${\displaystyle {}\exp x>1}$, and for ${\displaystyle {}x<0}$ we have ${\displaystyle {}\exp x<1}$.
6. The real exponential function is strictly increasing.

### Proof

(1) follows directly from the definition.
(2) follows from

${\displaystyle {}\exp x\cdot \exp {\left(-x\right)}=\exp {\left(x-x\right)}=\exp 0=1\,}$

using fact.
(3) follows for ${\displaystyle {}n\in \mathbb {N} }$ from fact by induction, and from that it follows with the help of (2) also for negative ${\displaystyle {}n}$.
(4). Nonnegativity follows from

${\displaystyle {}\exp x=\exp {\left({\frac {x}{2}}+{\frac {x}{2}}\right)}=\exp {\frac {x}{2}}\cdot \exp {\frac {x}{2}}={\left(\exp {\frac {x}{2}}\right)}^{2}\geq 0\,.}$

(5). For real ${\displaystyle {}x}$ we have ${\displaystyle {}\exp x\cdot \exp {\left(-x\right)}=1}$, so that because of (4), one factor must be ${\displaystyle {}\geq 1}$ and the other factor must be ${\displaystyle {}\leq 1}$. For ${\displaystyle {}x>0}$, we have

${\displaystyle {}\exp x=\sum _{n=0}^{\infty }{\frac {1}{n!}}x^{n}=1+x+{\frac {1}{2}}x^{2}+\ldots >1\,}$

as only positive numbers are added.
(6). For real ${\displaystyle {}y>x}$, we have ${\displaystyle {}y-x>0}$, and therefore, because of (5) ${\displaystyle {}\exp {\left(y-x\right)}>1}$, hence

${\displaystyle {}\exp y=\exp {\left(y-x+x\right)}=\exp {\left(y-x\right)}\cdot \exp x>\exp x\,.}$

${\displaystyle \Box }$