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Transcript
DEFINITION OF A SEQUENCE
A sequence is a set of numbers u1; u2; u3; . . . in a definite order of arrangement (i.e., a correspondence
with the natural numbers) and formed according to a definite rule.
Each number in the sequence is
called a term; unis called the nth term. The sequence is called finite or infinite according as there are or
are not a finite number of terms.
The sequence u1; u2; u3; . . . is also designated briefly by fung.
1. The set of numbers 2; 7; 12; 17; . . . ; 32 is a finite sequence; the
EXAMPLES.
un¼ 2 5ًn
1‫ ¼ ق‬5n
nth term is given by
3, n ¼ 1; 2; . . . ; 7.
2. The set of numbers 1; 1=3; 1=5; 1=7; . . . is an infinite sequence with
n ¼ 1; 2; 3; . . . .
nth term
un¼ 1=ً2n
1‫ق‬,
Unless otherwise specified, we shall consider infinite sequences only.
LIMIT OF A SEQUENCE
A number l is called the limit of an infinite sequence u1; u2; u3; . . . if for any positive number
find a positive number N depending on
we can
such that ju lj < for all integers n > N. In such case we
n
write lim un¼ l.
n!1
EXAMPLE .
If un¼ 3 1=n ¼ 3n 1‫=ق‬n, the sequence is 4; 7=2; 10=3; . . . and we can show that lim u n¼ 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 definition is applied to see if the guess is
really correct.
THEOREMS ON LIMITS OF SEQUENCES
If lim an¼ A and lim
n!1
1.
lim ًan bn‫¼ ق‬
n!1
2.
lim an
lim bn¼ A B
n!1
lim ًanbn‫¼ ق‬
n!1
3.
bn¼ B, then
n!1
n!1
lim anlim
n!1
bn¼ A
lim ًanbn ¼ ‫ ق‬lim an‫ ق‬lim
n!1
n!1
B
n!1
bn‫ ¼ ق‬AB
n!1
23
Copyright 2002, 1963 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.
24
SEQUENCES
4.
lim an A
¼
lim an¼ n!1
if
lim
n!1bn
B
n!1 bn
lim bn
[CHAP. 2
¼B6¼ 0
n!1
an does not exist.
If B ¼ 0 and A 6¼ 0, lim
n!1 bn
If B ¼ 0 and A ¼ 0, lim
5.
6.
n!1 bnmay
lim ap
n!1
an
n¼ lim
n!1
lim pa ¼ plim n!1
n
an‫ ق‬p¼Ap,
or may not exist.
for p ¼ any real number if Apexists.
an
¼ pA,
for p ¼ any real number if p Aexists.
n!1
INFINITY
We write lim an¼ 1 if for each positive number M we can find a positive number N (depending on
an!1
M) such thatn> M for all n > N. Similarly, we write lim
an¼ 1 if for each positive number M we
n!1
can find a positive number N such that an<
M for all n > N.
It should be emphasized that 1 and
1
are not numbers and the sequences are not convergent.
The terminology employed merely
indicates that the sequences diverge in a certain manner.
That is, no matter how large a number in
absolute value that one chooses there is an
n such that the absolute value of a is greater than that
n
quantity.
BOUNDED, MONOTONIC SEQUENCES
If u @ M for n ¼ 1; 2; 3; . . . ; where M is a constant (independent of n), we say that the sequence
n
fu g is bounded above and M is called an upper bound. If u A m, the sequence is bounded below and m is
n
n
called a lower bound.
If m @ u @ M
the sequence is called
n
bounded.
Often this is indicated by
ju j @ P.
n
Every
convergent sequence is bounded, but the converse is not necessarily true.
If un1Aunthe sequence is called monotonic increasing; if un1>unit is called strictly increasing.
Similarly, if
u [email protected] the sequence is called monotonic decreasing, while if
n
u 1<u it is strictly
n
n
n
decreasing.
EXAMPLES.
1. The sequence 1; 1:1; 1:11; 1:111; . . .
increasing.
2. The sequence 1;
3. The sequence
1; 1;
1;
is bounded and monotonic increasing.
It is also strictly
1; 1; . . . is bounded but not monotonic increasing or decreasing.
1:5;
2;
2:5;
3; . . . is monotonic decreasing and not bounded. However, it
is bounded above.
The following theorem is fundamental and is related to the Bolzano–Weierstrass theorem (Chapter
1, Page 6) which is proved in Problem 2.23.
Theorem.
Every bounded monotonic (increasing or decreasing) sequence has a limit.
LEAST UPPER BOUND AND GREATEST LOWER BOUND OF A SEQUENCE
A number
M is called the least upper bound (l.u.b.) of the sequence fu g if u @ M, n ¼ 1; 2; 3; . . .
while at least one term is greater than M
n
n
for any > 0.
A number m is called the greatest lower bound (g.l.b.) of the sequence fu g if u A m, n ¼ 1; 2; 3; . . .
n
n
while at least one term is less than m
for any > 0.
Compare with the definition of l.u.b. and g.l.b. for sets of numbers in general (see Page 6