1549901369-Elements_of_Real_Analysis__Denlinger_

436 Chapter 7 • The Riemann Integral

Now, for 0 <: c < 1,

1

1

)xdx= [2v'x]~=2(1-y'c).

Thus, lim j

1

~ dx = 2, so f 1 ~ dx converges, and f 1 ~ dx = 2. D

c->O+ c yX lo yX lo yX

Theorem 7.8.5 (The Comparison Test, I) Suppose a < b, and for all

a< x < b, 0:::; J(x):::; g(x), and f , g are integrable over [a,x]. If J: g converges,

then so does J: f, and J: f:::; J: g. (Of course, if J: f diverges, then so does

1: g.)

Proof. Suppose a < b, and Va < x < b, 0 :::; f(x) :::; g(x), and f , g are

integrable over [a, x]. Suppose J: g converges. Since g(x) 2 0 on [a, b], the

function G(x) = J: g is monotone increasing on [a, b). Since J: g converges,

lim J: g exists and

X->b-

J: g = limb_ J: g = sup { J: g : a < x < b}.

X->

(See Theorem 5.2.17 and Exercise 5.2.18.)

Similarly, the function F(x) = J: f is monotone increasing on [a, b) and
Vx E [a, b),

fax f:::; J: g:::; J: g (see above).

Thus, by Theorem 5.2.17, lim r f exists and is :::; lab g. That is, lab f

X->b- a

converges and J: f :::; J: g. •

Remark 7.8.6 The comparison test remains true if we replace "O ::; f(x) ::;

g(x)" by "O 2 f(x) 2 g(x)" or replace "[a, x]" by "[x, b]."

1

(^1 1 1 1)
Example 7.8.7 - 2 --dx converges because Vx E (0, 1], 0 :S - 2 --:S 2
0 x +x x +x x
and f
1
~ dx converges. D
lo x
Theorem 7.8.8 (Absolute Convergence) Suppose J: f is an improper in-
tegral of Type I. If J: lfl converges, then so does J: f. [In this case, we say that
J: f converges absolutely.]