1549901369-Elements_of_Real_Analysis__Denlinger_

466 Chapter 8 • Infinite Series of Real Numbers

00 00
Apply the comparison test to the series L an and L an 0 rn. Since the
n=no+l n=l
second series is a geometric series with 0 < r < 1, it converges. Therefore,
00
the above inequalities together with the comparison test show that L an

00
converges. Therefore, L an converges.
n=l

n=no+l

(b) Suppose an+l 2: 1 for all but finitely many n. That is , 3 no E N 3

an

n 2: no :::;. --an+l 2: 1 :::;> an+l 2: an. Then n 2: no :::;. an 2: an
an^0 > 0. But then
00
an~ o. Therefore, by the general term test, L an diverges. •
n=l

Theorem 8.2.10 states the ratio test in basic form. However, the student
may remember the ratio test by the more familiar limit form in which it ap-
pears in elementary calculus. The familiar form is frequently preferred since it
suggests a direct procedure that can be used whenever a certain limit exists
and is not l. It is easily proved as a corollary to Theorem 8.2.10.

Corollary 8.2.11 (Ratio Test, Limit Form) If Lan is a nonnegative

series and lim an+l = L (possibly +oo) then

n~oo an

(a) if L < 1, the series Lan converges, and

(b) if L > 1, the series Lan diverges.

Proof. Exercise 36. •

Note that the ratio test gives us no information about the convergence or
divergence of Lan when L = 1. It is customary to say that the ratio test "fails"
when L = 1, although it would be more correct to say that it is "inconclusive"
in this case. There are nonnegative series Lan with L = 1 that converge and
others that diverge. For example, L = 1 for both L ~ and L ~, yet the former
diverges and the latter converges.

Examples 8.2.12 Use the ratio test to test the following series for convergence
or divergence:
oo n2 + 1
(a) I:~
n=l

oo I

(b)" ,L_,, ~ 3n

n=l

00

(c)Ln3:1

n=l