Figure 1 example 4 (a) We will be interested in comparing the growth rates of familiar

sequences like log₂ n, , n²,2ⁿ, n! and nⁿ. Even for relatively small values of n it seems clear from Figure 1 that nⁿ grows a lot faster than n!, which grows a lot faster than 2ⁿ, etc., although log₂ n and seem to be running close to each other. In § 1.6 we will make these ideas more precise and give arguments that don’t rely on appearances based on a few calculations.

(b) We are really interested in comparing the sequences in part (a) for

large values of n. See Figure 2.¹¹ It now appears that log₂ n does grow [a lot] slower than , a fact that we will verify in the next section. The growth is slower because 2ⁿ grows [a lot] faster than n², and log₂ x and are the inverse functions of 2^x and x², respectively. ■

		n	n2	2n	n!	nn
3.32	3.16	10	100	1,024	3.63 • 106	1010
6.64	10	100	10,000	1.27-1030	9.33 • 10157	10200
9.97	31.62	1,000	106	1.07.10301	4.02 •102567	103000
13.29	100	10,000	108	2.00 •103010	2.85 •1035659	1040000
16.61	316.2	100,000	1010	1.00.1030103	2.82 • 10456573	10500000
19.93	1000	106	1012	9.90 • 10301029	8.26 • l05565708	l06000000
39.86	106	1012	1024	big	bigger	biggest

FIGURE 2

So far, all of our sequences have had real numbers as values. How­ever, there is no such restriction in the definition and, in fact, we will be interested in sequences with values of other sorts.

example 5 The following sequences have values that are sets.

(a) A sequence (D_n)_n of subsets of Z is defined by

D_n = {m  Z: m is a multiple of n} ={0, ±n, ±2n, ±3n, ...}.

(b) Let  be an alphabet. For each k  N,  ^k is defined to be the set of all words in ^* having length k. In symbols,

 ^k = {w  ^*: length(w) = k}.

The sequence ( ^k)_k is a sequence of subsets of ^* whose union,

is ^*. Note that the sets  ^k are disjoint, that ⁰ = {}, and that ¹ = . In case  = {a, b}, we have ⁰ = {}, ¹ =  = {a, b}, ² = {aa, ab, ba, bb], etc.

The sets ^k appeared briefly in Example 7 of § 1.4 with the names L^(k) where L was the length function on *. Henceforth we will use the notation ^k for these sets. ■

In the last example we wrote for the union of an infinite sequence of sets. A word is a member of this set if it belongs to one of the sets ^k. The sets ^k are disjoint, but in general we consider unions of sets that may overlap. To be specific, if (A _k)_k is a sequence of sets, then we define

This definition makes sense, of course, if the sets are defined for k in P or in some other set. Similarly, we define

.

The notation has a similar interpretation except that  plays a special role. The notation signifies that k takes the values 0,1, 2, ... without stopping; but k does not take the value . Thus

whereas

.

In Example 5 we could just as well have written ^* = .

Some lists are not infinite. A finite sequence is a string of objects that are listed using subscripts from a finite subset of Z of the form {m, m + 1, ..., n}. Frequently, m will be 0 or 1. Such a sequence is a function with domain {m, m + 1, ..., n}, just as an infinite sequence has domain {m, m + 1,...}.