Functions defined on the set of integers larger than a given integer are called sequences.
Definition 4.2.1 A function whose domain is defined as a set of the form
for k an integer is known as a sequence. Thus you can consider f
etc. Usually the domain of the sequence is either ℕ, the natural numbers consisting of
or the nonnegative integers,
. Also, it is traditional to
write f1,f2, etc. instead of f
etc. when referring to sequences. In
the above context, fk is called the first term, fk+1 the second and so forth. It is
also common to write the sequence, not as f but as
i=k∞ or just
Example 4.2.2 Let
k=1∞ be defined by ak ≡ k2
This gives a sequence. In fact, a7 = a
+ 1 = 50 just from using the formula for the
term of the sequence.
It is nice when sequences come in this way from a formula for the kth 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 an+1 from knowledge of a1,
This rule which
from knowledge of ak
for k ≤ n
is known as a recurrence relation.
Example 4.2.3 Let a1 = 1 and a2 = 1. Assuming a1,
,an+1 are known, an+2 ≡
Thus the first several terms of this sequence, listed in order, are 1, 1, 2, 3, 5, 8,
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 4.2.4 Let
be a sequence and let n1 < n2 < n3,
strictly increasing list of integers such that n1 is at least as large as the first number in
the domain of the function. Then if bk ≡ ank,
is called a subsequence of
For example, suppose an =
= 2, a3
then letting bk = ank, it follows