A sequence is a set of numbers u1; u2; u3; . . . in a de?nite order of arrangement (i.e., a correspondence
with the natural numbers) and formed according to a de?nite rule. Each number in the sequence is
called a term; unis called the nth term. The sequence is called ?nite or in?nite according as there are or
are not a ?nite number of terms. The sequence u1; u2; u3; . . . is also designated brie?y by fung.
EXAMPLES. 1. The set of numbers 2; 7; 12; 17; . . . ; 32 is a ?nite sequence; the nth term is given by
un¼ 2 5ًn 1ق ¼ 5n 3, n ¼ 1; 2; . . . ; 7.
2. The set of numbers 1; 1=3; 1=5; 1=7; . . . is an in?nite sequence with nth term un¼ 1=ً2n 1ق,
n ¼ 1; 2; 3; . . . .
Unless otherwise speci?ed, we shall consider in?nite sequences only.
LIMIT OF A SEQUENCE
A number l is called the limit of an in?nite sequence u1; u2; u3; . . . if for any positive number we can
?nd a positive number N depending on such that junlj < for all integers n > N. In such case we
write lim un¼ l.
n!1
EXAMPLE . If un¼ 3 1=n ¼ ً3n 1ق=n, the sequence is 4; 7=2; 10=3; . . . and we can show that lim un¼ 3.
n!1
If the limit of a sequence exists, the sequence is called convergent; otherwise, it is called divergent. A
sequence can converge to only one limit, i.e., if a limit exists, it is unique. See Problem 2.8.
A more intuitive but unrigorous way of expressing this concept of limit is to say that a sequence
u1; u2; u3; . . . has a limit l if the successive terms get ‘‘closer and closer’’ to l. This is often used to
provide a ‘‘guess’’ as to the value of the limit, after which the de?nition is applied to see if the guess is
really correct.