a. Show that for any integer
b. Show that for any integer .
c. Sketch a graph to illustrate the results of parts (a) and (b).
d. Use the O- and -notations to express the results of parts (a) and (b).
e. What can you deduce about the order of ?
The function is greater than or equal and less than or equal for .
The functions and are defined on and all are positive integers where .
It is clear that for any integer , is greater than or equal zero.
As an example,
Therefore, for any integer .
Also, when , as we proved previously.
Let’s assume that for any integer where .
The function is greater than or equal for .
The relationships we have obtained in part (a.) and part (b.)
( and )
The relationship between the functions , and in the notations of and .
The order of the functions .
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