elementary proof of growth of exponential function
If is a non-negative real number and is a non-negative integer, then .
If is a real number such that and and are non-negative integers, we have .
Let . Write where and are non-negative integers and .
By the preceding proposition, . Raising both sides of this inequality to the power, we have . Since , we also have ; multiplying both sides by this inequality and collecting terms,
Multiplying the right-hand side by and rearranging,
Since , we also have
Recalling that and , we conclude that
If , , and are real numbers such that , and , then
Let and be integers such that and . Since , we have . By the preceeding proposition, we have
Since , we have , so
Since , we have
Summarrizing our progress so far,
Dividing both sides by and simplifying,
If and are real numbers and , then
Dividing by and rearranging,
Since and , we also have by the squeeze rule.
|Title||elementary proof of growth of exponential function|
|Date of creation||2014-03-10 17:57:26|
|Last modified on||2014-03-10 17:57:26|
|Last modified by||rspuzio (6075)|