Page 33 - Schaum's Outline of Theory and Problems of Advanced Calculus

24 SEQUENCES [CHAP. 2

lim a n
a n A
4. lim ¼ n!1 ¼ if lim b n ¼ B 6¼ 0
n!1 b n lim b n B n!1
n!1
a n
If B ¼ 0 and A 6¼ 0, lim does not exist.
n!1 b n
a n
If B ¼ 0 and A ¼ 0, lim may or may not exist.
n!1 b n
p
p
p
p
5. lim a n ¼ð lim a n Þ ¼ A , for p ¼ any real number if A exists.
n!1 n!1
lim a n A A
6. lim p ¼ p n!1 ¼ p , for p ¼ any real number if p exists.
a n
n!1
INFINITY
We write lim a n ¼1 if for each positive number M we can find a positive number N (depending on
M)such that a n > M for all n > N. Similarly, we write lim a n ¼ 1 if for each positive number M we
n!1
n!1
can find a positive number N such that a n < 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 n is greater than that
quantity.

BOUNDED, MONOTONIC SEQUENCES
If u n @ M for n ¼ 1; 2; 3; .. . ; where M is a constant (independent of n), we say that the sequence
fu n g is bounded above and M is called an upper bound.If u n A m, the sequence is bounded below and m is
called a lower bound.
If m @ u n @ M the sequence is called bounded. Often this is indicated by ju n j @ P. Every
convergent sequence is bounded, but the converse is not necessarily true.
If u nþ1 A u n the sequence is called monotonic increasing;if u nþ1 > u n it is called strictly increasing.
Similarly, if u nþ1 @ u n the sequence is called monotonic decreasing, while if u nþ1 < u n it is strictly
decreasing.

EXAMPLES. 1. The sequence 1; 1:1; 1:11; 1:111; .. . is bounded and monotonic increasing. It is also strictly
increasing.
2. The sequence 1; 1; 1; 1; 1; ... is bounded but not monotonic increasing or decreasing.
3. The sequence 1; 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 n g if u n @ M, n ¼ 1; 2; 3; ...
while at least one term is greater than M for any > 0.
m
m
A number m is called the greatest lower bound (g.l.b.) of the sequence fu n g if u n A m, n ¼ 1; 2; 3; ...
while at least one term is less than m þ for any > 0.
m
Compare with the definition of l.u.b. and g.l.b. for sets of numbers in general (see Page 6).