1549901369-Elements_of_Real_Analysis__Denlinger_

86 Chapter 2 • Sequences

n > no :::::} an > M and Cn < -1

:::::} an > M and - Cn > 1

:::::} (an)(-Cn) > M

:::::} -anCn > M

:::::} anCn < -M.

Therefore, lim (ancn) = -oo. •
n-.oo
Symbolic Shorthand: The results of Theorem 2.4.9 are often expressed
as a kind of "algebra" of +oo and -oo, summarized in Table 2.1 as follows:

Table 2.1

Algebra of Infinite Limits

(a) (+oo) + (+oo) = +oo

(b) (-oo)+(-oo)=-oo

( c) ( +oo) · ( + oo) = +oo

(d) (-oo) · (-oo) = +oo

(e) (+oo) · (-oo) = -oo

However, the forms ( +oo) + ( - oo) and ( +oo) - ( + oo) are "indeterminate"
in the sense that no answer can be given that is always true. That is , there are
pairs of sequences {an} and {bn} such that lim an= +oo and lim bn = -oo
n-+oo n---+oo
for which n---+oo lim (an +bn) = +oo, others for which n---+oo lim (an +bn) = -oo, others for

which lim (an+ bn) is a finite number, and still others for which lim (an+ bn)
n---+oo n---+oo
does not exist.
In the same sense (of limits of sequences), we can combine finite and infinite
limits algebraically. Suppose p > 0 and n < 0 represent finite positive and
negative limits of sequences. Table 2.2 summarizes the results:

Table 2.2

Algebra of Infinite Limits

(a) (+oo) + p(or n) = +oo

(b) (-oo) + p(or n) = -oo

(c) (±oo) · p = ±oo

(d) (±oo) · n = =t=oo

(e)

(f)

(g)

( ±oo) · 0 is indeterminate

_1_ =0

±oo

1 1

IOJ = + oo, but

0

is indeterminate