5. Sequences

This section is concerned with lists of things. Subscript notation comes in handy in this context and whenever we deal with large collections of objects; here “large” often means “more than 3 or 4.” For example, letters x, y and z are adequate when dealing with equations involving three or fewer unknowns. But if there are ten unknowns, or if we wish to discuss the general situation of n unknowns for some unspecified integer n in P, then would be a good choice for the names of the unknowns. Here we distinguish the unknowns by the little numbers 1,2, ..., n written below the x’s, which are called subscripts. As another example, a general nonzero polynomial has the form

where a_n  0. Here n is the degree of the polynomial and the n + 1 pos­sible coefficients are labeled using subscripts. For example, the polynomial x³ + 4x² - 73 fits this general scheme, with n = 3, a₃ = 1, a₂ = 4, a₁ = 0 and a₀ = -73.

We have used the symbol  as a name for an alphabet. In mathe­matics the big Greek sigma  has a standard use as a general summation sign. The terms following it are to be summed according to how  is decorated. For example, consider the expression

The decorations “k = 1” and “10” tell us to add up the numbers k² obtained by successively setting k = 1, then k = 2, then k = 3, etc. on up to k = 10. That is,

= 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 + 81 +100 = 385.

The letter k is a variable [it varies from 1 to 10] that could be replaced here by any other variable. Thus

^{= =}

We can also consider more general sums like

in which the stopping point n can take on different values. Each value of n gives a particular value of the sum; for each choice of n the variable k takes on the values from 1 to n. Here are some of the sums represented by

Value of n The sum

n = l 1²=1

n = 2 1² + 2² = l + 4 = 5

n = 3 1² + 2² + 3² = 14

n = 4 1² + 2² + 3² +4² = 30

n = 10 1² + 2² + 3² + 4² + 5² + 6² + 7² + 8² + 9² + 10² = 385

n = 73 1² + 2² + 3² + 4² + … + 73² = 132,349

We can also discuss even more general sums such as

and .

Here it is understood that {x_k: 1  k  n} and {a_j: m j  n} represent collections of numbers. Presumably m  n, since otherwise there would be nothing to sum.

In analogy with , the big Greek pi  is a general product sign. For n  P the product of the first n integers is called n factorial and is written n!. Thus

The expression is somewhat confusing for small values of n like 1 and 2; it really means “multiply consecutive integers until you reach n.” The expression is less ambiguous. Here are the first few values of n!: 1! = 1, 2! = 1  2 = 2, 3! = 1  2  3 = 6, 4! = 1  2  3  4 = 24. More values of n! appear in Figures 1 and 2. For technical reasons n! is also defined for n = 0; 0! is defined to be 1. The definition of n! will , be reexamined in §4.1.

An infinite string of objects can be listed by using subscripts from the set N = {0, 1, 2, ...} of natural numbers [or {m, m + 1, m + 2,...} for some integer m]. Such strings are called sequences. Thus a sequence on N is a list that has a specified value s, for each integer n  N. We frequently call s, the nth term of the sequence. It is often convenient to denote the sequence itself by (s_n) or (s_n)_n or . Sometimes we will write s(n) instead of s_n. Computer scientists commonly use the notation s[n], in part because it is easy to type on a terminal.

The notation s(n) looks like our notation for functions. In fact, a sequence is a function whose domain is the set N = {0, 1, 2, ...} of natural numbers or is {m, m + 1, m + 2,...} for some integer m. Each integer n in the domain of the sequence determines the value s(n) of the nth term.

Our first examples will be sequences of real numbers.

example 1 (a) The sequence (s_n)_n where s_n = n! is just the sequence

(1, 1, 2, 6, 24, ...) of factorials. The set of values is {1, 2, 6, 24, ...} =

{n!: n  P}.

(b) The sequence (a_n)_n given by a_n = (-1)ⁿ for n  N is the se-

quence (1, - 1, 1, - 1, 1, - 1, 1, ...) whose set of values is {- 1,1}. ■

As the last example suggests, it is important to distinguish between a sequence and its set of values. We always use braces { } to list or describe a set and never use them to describe a sequence. The sequence (a_n)_n given by a_n = (-1)ⁿ in Example l(b) has an infinite number of terms, even though their values are repeated over and over. On the other hand, the set of values {(- l)ⁿ: n  N} is exactly the set {- 1,1} consisting of two numbers.

Sequences are frequently given suggestive abbreviated names, such as seq, fact, sum, and the like.

example 2 (a) Let fact(n) = n! for n  N. This is exactly the same sequence as in

Example l(a); only its name [fact, instead of s] has been changed. Note

that fact(n + 1) = (n + 1) * fact(n) for n  N, where * denotes

multiplication of integers.

(b) For n  N, let two(n) = 2ⁿ. Then two is a sequence. Note that

two(n + 1) = 2 * two(n) for n  N. ■

Our definition of sequence allows the domain to be any set of the form {m, m + 1, m + 2,...} where m is an integer.

EXAMPLE 3 (a) The sequence (b_n) given by b_n = 1/n² for n ≥ 1 clearly needs to have

its domain avoid the value n = 0. The first few values of the sequence

are .

(b) Consider the sequence whose nth term is log₂ n. Note that log₂ 0

makes no sense, so this sequence must begin with n = 1. We have

log₂ 1=0 since 2⁰ = 1, log₂ 2=1 since 2¹ = 2, log₂ 4=2 since 2² = 4,

log₂ 8=3 since 2³ == 8, etc. The intermediate values of log₂ n can only be

approximated. See Figure 1. For example, log₂ 5  2.3219 is only an ap-­

proximation since 2^2.3219  4.9999026. ■

		n	n2	2n	n!	nn
0	1.0000	1	1	2	1	1
1.0000	1.4142	2	4	4	2	4
1.5850	1.7321	3	9	8	6	27
2.0000	2.0000	4	16	16	24	256
2.3219	2.2361	5	25	32	120	3125
2.5850	2.4495	6	36	64	720	46,656
2.8074	2.6458	7	49	128	5040	823,543
3.0000	2.8284	8	64	256	40,320	1.67.107
3.1699	3.0000	9	81	512	362,880	3.87 • 108
3.3219	3.1623	10	100	1,024	3,628,800	1010