In this section we examine the asymptotic behavior of polynomials in *n*.
In particular, we will see that as *n* gets large,
the term involving the highest power of *n* will dominate all the others.
Therefore, the asymptotic behavior is determined by that term.

TheoremConsider a polynomial innof the form

where . Then .

extbfProof
Each of the terms in the summation is of the form .
Since *n* is non-negative,
a particular term will be negative only if .
Hence, for each term in the summation, .
Recall too that we have stipulated that the coefficient of the
largest power of *n* is positive, i.e., .

Note that for integers , for . Thus

From Equation we see that we have found the constants and , such that for all , . Thus, .

This property of the asymptotic behavior of polynomials
is used extensively.
In fact, whenever we have a function,
which is a polynomial in *n*,
we will immediately ``drop'' the less significant terms
(i.e., terms involving powers of *n* which are less than *m*),
as well as the leading coefficient, ,
to write .

Copyright © 1997 by Bruno R. Preiss, P.Eng. All rights reserved.