1549901369-Elements_of_Real_Analysis__Denlinger_

8.2 Nonnegative Series 469

Examples 8.2.17 Use the root test to test the following series for convergence

or divergence:

(a) f n22~ 1

n=l

oo I

(b) ~ !:!:.:

~3n

n=l

00

(c)Ln3:1

n=l

Solution. These examples are the same as those considered in Example 8.2.12.

We shall see here how using the root test differs from using the ratio test.

() a^1. im ~n2+1 ---= l' im V'n2+1 = 2 i1 im v n~N n-+ 1. ow,

n->oo 2n n->oo 2 n->oo

( yln)

2

= \/n2 < V'n2 + 1 < V'2rt2 = V2 ( yln)

2

. By Examples 2.3.8 and 2.3.9,
yin-+ 1 and V2-+ 1. Thus, by the squeeze theorem, V'n^2 + 1-+ 1. Therefore,

in the notation of Theorem 8.2.15, R = lim ~n

2

• 1
  = ~· Since L < 1, the
  n->oo 2n
  00 2+1
  limit form of the root test guarantees that ~ ~ _n __ 2n converges.
  n=l

(b) In the notation of Theorem 8.2.15, R = lim nfri! = lim Y'nf.
n->oo V ~ n->oo 3
oo I
Now, lim V'nf = +oo [see Exercise 2.6.22]. Thus, R = +oo. Therefore, ~ !!'.:
~00 ~~
n=l
diverges (to +oo).

(c) In the notation of Theorem 8.2.15, R = lim ~ 3 n. Now,

n->oo n + 1

~ ( Jn)2 = ff, < ~ n3: 1 < *' = (£ = ( Jn)^2 · Since
V2 -+ 1 and yin -+ 1, these inequalities along with the squeeze principle tell
us that R = 1, so we must use another test. As noted in Example 8.2.12 (c),

this series converges by the comparison test. D

The alert reader surely noticed that the ratio test and the root test pro-
duced identical limits (L = R) in all three series in Example 8.2.17. This does
not always happen, but it happens often enough for us to look for a reason. In
fact, if the ratio test proves that a nonnegative series converges (or diverges),
the root test will also. But the converse relation is not true. The following
theorem and example show what is going on.

Theorem 8.2.18 Given any sequence {an} of positive real numbers,

1 Im. --an+l < _ l' Im y nj;;-an _ < -1· Im - y nr,:;-an < _ -1· Im - --an+l.

n-+OO an n-+-OO n-+OO n-+OO an

Thus, if lim an+l exists, then lim ~ also exists and is equal to it.

n-+oo an n-+oo