Real exponential function via exponential series/Introduction/Section

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For every , the series

is called the exponential series in .

So this is just the series


For every , the exponential series


absolutely convergent.


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 graph of the real exponential function


The function

is called the (real)

exponential function.

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


For real numbers , the equation



The Cauchy product of the two exponential series is


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.


The exponential function

fulfills the following properties.
  1. .
  2. For every , we have . In particular .
  3. For integers , the relation holds.
  4. For every , we have .
  5. For we have , and for we have .
  6. 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