In this chapter, we study the rate of growth of positive sequences. We introduce a formal definition that enables us to say that one sequence does not grow faster than another sequence. Suppose we have two sequences and. We could say that xi does not grow faster than yi if xi ≤ yi for every i. However, such a restricted definition is rather limited, as suggested by the following examples:

1. The sequence xi is constant: xi = 1000 for every i, while the sequence yi is defined by yi = i. Clearly we would like to say that yi grows faster than xi even though y100 < x100.

2. The sequences satisfy xi = yi + 5 or xi = 2 · yi for every i. In this case, we would like to say that the two sequences grow at the same rate even though xi > yi.

Thus we are looking for a definition that is insensitive to the values of finite prefixes of the sequence. In addition, we are looking for a definition that is insensitive to addition or multiplication by constants. This definition is called the asymptotic behavior of a sequence.

ORDER OF GROWTH RATES

Consider the Fibonacci sequence. The exact value of g(n), or an analytic equation for g(n), is interesting but sometimes all we need to know is how fast does g(n)grow?