# Real exponential function via exponential series/Introduction/Section

So this is just the series

For , the statement is clear. Else, we consider the fraction

This is, for , smaller than . By the ratio test, we get convergence.

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

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

The Cauchy product of the two exponential series is

where

This series is due to fact absolutely convergent and the limit is the product of the two limits. Furthermore, the -th summand of the exponential series of equals

so that both sides coincide.

- .
- For every , we have . In particular .
- For integers , the relation holds.
- For every , we have .
- For we have , and for we have .
- The real exponential function is strictly increasing.

(1) follows directly from the definition.

(2) follows from

using
fact.

(3) follows for
from
fact
by induction, and from that it follows with the help of (2) also for negative .

(4). Nonnegativity follows from

(5). For real we have , so that because of (4), one factor must be and the other factor must be . For , we have

as only positive numbers are added.

(6). For real
,
we have
,
and therefore, because of (5)
,
hence