exponential function defined as limit of powers
It is possible to define the exponential function and the natural logarithm in terms of a limit of powers. In this entry, we shall present these definitions after some background information and demonstrate the basic properties of these functions from these definitions.
Two basic results which are needed to make this development possible are the following:
Suppose that is a sequence such that . Then ,
For proofs, see the attachments. From them, we first conclude that a sequence converges.
Let be any real number. Then the sequence
The foregoing results show that the limit in the following definition converges, and hence defines a bona fide function.
Let be a real number. Then we define
We may now derive some of the chief properties of this function. starting with the addition formula.
For any two real numbers and , we have .
The function is strictly increasing.
Suppose that is a strictly positive real number. By theorem 1 and the definition of the exponential as a limit, we have , so we conclude that implies .
Now, suppose that and are two real numbers with . Since , we have . Using theorem 4, we have , so the function is strictly increasing. ∎
The function is continuous.
Suppose that . By theorem 1 and the definition of the exponential as a limit, we have and . By theorem 4, . Hence, we have the bounds and . From the former bound, we conclude that and, from the latter, that , so .
Suppose that is any real number. By theorem 4, . Hence, . In other words, for all real , we have , so the exponential function is continuous. ∎
The function is one-to-one and maps onto the positive real axis.
The one-to-one property follows readily from monotonicity — if , then we must have , because otherwise, either or , which would imply or , respectively. Next, suppose that is a real number greater than . By theorem 1 and the definition of the exponential as a limit, we have . Thus, ; since is continuous, the intermediate value theorem asserts that there must exist a real number between and such that . If, instead, , then so we have a real number such that . By theorem 4, we then have . So, given any real number , there exists a real number such that , hence the function maps onto the positive real axis. ∎
The function is convex.
Since the function is already known to be continuous, it suffices to show that for all real numbers and . Changing variables, this is equivalent to showing that for all real numbers and . By theorem 4, we have
Using the inequality with and multiplying by , we conclude that , hence the exponential function is convex. ∎
Defining the constant as , we find that the exponential function gives powers of this number.
For every real number , we have .
|Title||exponential function defined as limit of powers|
|Date of creation||2013-03-22 17:01:35|
|Last modified on||2013-03-22 17:01:35|
|Last modified by||rspuzio (6075)|