1549901369-Elements_of_Real_Analysis__Denlinger_

462 Chapter 8 m Infinite Series of Real Numbers

00 00
Thus, by the integral test, '""" n diverges. Therefore, '""" n di-
n=4 ~ n2+10 n=l ~n^2 +10
verges. D

Corollary 8.2.5 (p-series) Let p be a fixed real number. The p-series
00 1
L P converges if p > 1 and diverges if p :::; 1.
n=l n

Proof. Let p be a fixed real number.
1 00 1
Case 1 (p :::; 0): By Theorem 5.6.15, - f+ 0, so L - diverges by the
nP n=l nP
general term test.
1
Case 2 (p > 0): The function f(x) = - is continuous and monotone
xP

decreasing on [1,oo), and lim 2._ = 0, by Theorem 5.6.15. Hence, we can
x-.oo xP
00 1
apply the integral test to the series L -.
n=l nP
Subcase 2a (p =f. 1) : Then

J

oo -d (^1) x = lim J b x - Pdx
1 xP b->oo 1
1

. x
[

1-p] b

= im --

b->oo 1 - p 1

= --^1 [. hm b^1 - P - 1. ]

1 - p b->oo

(7)

If p > 1, lim b^1 -P = 0 by Theorem 5.6.15. Then by (7), -dx con-

J

oo 1
hoo 1 ~
00 1
verges, so by the integral test, L - converges.
n=l nP
If p < 1, lim b^1 -P = +oo by Theorem 5.6.14. Then by (7), -dx
J

oo 1

hoo 1 ~

diverges, so by the integral test, f: 2._ diverges.
n=l nP
Subcase 2b (p = 1): Then

J

oo -d (^1) x = lim Jb -dx 1
1 xP b-.oo 1 x
= lim lnb = +oo.
b->oo
J
oo 1 00 1
In this case, -dx diverges, so by the integral test, L - diverges.
1 ~ ~1~ •