1549901369-Elements_of_Real_Analysis__Denlinger_

464 Chapter 8 • Infinite Series of Real Numbers

Theorem 8.2.8 (Limit Comparison Test) Suppose Lan and L bn are

nonnegative series. Let lim abn = p (possibly +oo).

n-+oo n

(a) If 0 < p < oo, then either both series converge or both series diverge.

(b) If p = 0 and L bn converges, then Lan converges.

(Equivalently, if p = 0 and Lan diverges, then L bn diverges.)

(c) If p = +oo and L bn diverges, then Lan diverges.

(Equivalently, if p = +oo and Lan converges, then L bn converges.)

a

Proof. Suppose Lan and L bn are nonnegative series, and let lim bn = p

n-+oo n

(possibly +oo). The three statements of this theorem can be proved in two

parts, as follows:

Part 1 (0 ::; p < oo, and L bn converges): Since { ~=} converges to a finite

number, it is bounded. Thus, ?JM> 0 3 Vn EN, ~= ::; M. Then

(a) Vn E N, an ::; Mbn, and

(b) L Mbn converges.

Thus, by the comparison test, Lan converges.

a

Part 2 (0 < p ::; oo, and L bn diverges): Since lim bn > 0, ?JM > 0 and

n-+oo n

an

?Jno EN 3 n;::: no=> bn > M. Then,

(a) n;::: no=> an> Mbn, and

(b) L Mbn diverges.

Thus, by the comparison test, Lan diverges. •

Examples 8.2.9 Use the limit comparison test to prove that the following
series converge (or diverge):

(a) f v'5n - 10

n=l 3n+ Vn

Solution.

(b) ~ lnn

~ n2

n=l

(a) As n gets large, the terms of the series (a) are something like Vn =

1 n

Vn. Thus, we will try to show that this series diverges by comparing it with

L Jn, which is a divergent p-series (p = ~). Let an = ~ -; and bn =