Functions defined on the set of integers larger than a given integer are called sequences.

Definition 4.2.1A function whosedomain is defined as a set of the form

{k,k + 1,k + 2,⋅⋅⋅}

for k an integer is known as a sequence. Thus you can consider f

(k)

,f

(k +1)

,f

(k + 2)

,etc. Usually the domain of the sequence is either ℕ, the natural numbers consisting of

{1,2,3,⋅⋅⋅}

or the nonnegative integers,

{0,1,2,3,⋅⋅⋅}

. Also, it is traditional towrite f_{1},f_{2}, etc. instead of f

(1)

,f

(2)

,f

(3)

etc. when referring to sequences. Inthe above context, f_{k}is called the first term, f_{k+1}the second and so forth. It isalso common to write the sequence, not as f but as

{fi}

_{i=k}^{∞}or just

{fi}

forshort.

Example 4.2.2Let

{ak}

_{k=1}^{∞}be defined by a_{k}≡ k^{2} + 1.

This gives a sequence. In fact, a_{7} = a

(7)

= 7^{2} + 1 = 50 just from using the formula for the
k^{th} term of the sequence.

It is nice when sequences come in this way from a formula for the k^{th} term.
However, this is often not the case. Sometimes sequences are defined recursively. This
happens, when the first several terms of the sequence are given and then a rule is
specified which determines a_{n+1} from knowledge of a_{1},

⋅⋅⋅

,a_{n}. This rule which
specifies a_{n+1} from knowledge of a_{k} for k ≤ n is known as a recurrence relation.

Example 4.2.3Let a_{1} = 1 and a_{2} = 1. Assuming a_{1},

⋅⋅⋅

,a_{n+1}are known, a_{n+2}≡a_{n} + a_{n+1}.

Thus the first several terms of this sequence, listed in order, are 1, 1, 2, 3, 5, 8,

⋅⋅⋅

. This
particular sequence is called the Fibonacci sequence and is important in the study of
reproducing rabbits. Note this defines a function without giving a formula for it. Such
sequences occur naturally in the solution of differential equations using power series methods
and in many other situations of great importance.

For sequences, it is very important to consider something called a subsequence.

Definition 4.2.4Let

{an}

be a sequence and let n_{1}< n_{2}< n_{3},

⋅⋅⋅

be anystrictly increasing list of integers such that n_{1}is at least as large as the first number inthe domain of the function. Then if b_{k}≡ a_{nk},