Page 321 - Schaum's Outline of Theory and Problems of Advanced Calculus

312 IMPROPER INTEGRALS [CHAP. 12

ð b ð b
(b) If A ¼ 0in (a), then gðxÞ dx converges, then f ðxÞ dx converges.
a a
ð b ð b
(c) If A ¼1 in (a), and gðxÞ dx diverges, then f ðxÞ dx diverges.
a a
This test is related to the comparison test and is a very useful alternative to it. In particular
p
taking gðxÞ¼ 1=ðx aÞ we have from known facts about the p integral the following theorems.
p
Theorem 3. Let lim ðx aÞ f ðxÞ¼ A. Then
x!aþ
ð b
(i) f ðxÞ dx converges if p < 1and A is finite
a
ð b
(ii) f ðxÞ dx diverges if p A 1 and A 6¼ 0(A may be infinite).
a
If f ðxÞ becomes unbounded only at the upper limit these conditions are replaced by those in

p
Theorem 4. Let lim ðb xÞ f ðxÞ¼ B. Then
x!b
b
ð
(i) f ðxÞ dx converges if p < 1and B is finite
a
b
ð
(ii) f ðxÞ dx diverges if p A 1 and B 6¼ 0(B may be infinite).
a
5 dx 1 x 1 1
ð r ffiffiffiffiffiffiffiffiffiffiffiffiffi
EXAMPLE 1. p ffiffiffiffiffiffiffiffiffiffiffiffiffi converges, since lim ðx 1Þ 1=2 1=2 ¼ lim 4 ¼ .
4
4
1 x 1 x!1þ ðx 1Þ x!1þ x 1 2
3 dx 1 1
ð
EXAMPLE 2. p ffiffiffiffiffiffiffiffiffiffiffiffiffi diverges, since lim ð3 xÞ p ffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ p ffiffiffiffiffi.
2
2
0 ð3 xÞ x þ 1 x!3 ð3 xÞ x þ 1 10
ð b ð b
3. Absolute and conditional convergence. f ðxÞ dx is called absolute convergent if j f ðxÞj dx
a a
ð b ð b ð b
converges. If f ðxÞ dx converges but j f ðxÞj dx diverges, then f ðxÞ dx is called condition-
a a a
ally convergent.
ð b ð b
Theorem 5. If j f ðxÞj dx converges, then f ðxÞ dx converges. In words, an absolutely convergent
a a
integral converges.

1 dx
ð 4
1
sin x
EXAMPLE. ffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffi and ffiffiffiffiffiffiffiffiffiffiffiffi converges ( p integral with a ¼ ; p ¼ ), it follows that
x x x
Since p
3
3
3 @ p p 3
4 ð 4
ð
sin x
sin x
ffiffiffiffiffiffiffiffiffiffiffiffi dx converges and thus ffiffiffiffiffiffiffiffiffiffiffiffi dx converges (absolutely).
p p
3
3 x x
Any of the tests used for integrals with non-negative integrands can be used to test for absolute
convergence.