Page 318 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 12] IMPROPER INTEGRALS 309

2. Quotient test for integrals with non-negative integrands.
ð ð
1 1
(a)If f ðxÞ A 0 and gðxÞ A 0, and if lim f ðxÞ ¼ A 6¼ 0or 1, then f ðxÞ dx and gðxÞ dx
a a
x!1 gðxÞ
either both converge or both diverge.
ð ð
1 1
(b)If A ¼ 0in(a)and gðxÞ dx converges, then f ðxÞ dx converges.
a a
ð ð
1 1
(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 often a very useful alternative to it. In particular,
p
taking gðxÞ¼ 1=x ,we have from known facts about the p integral, the following theorem.
p
Theorem 1. Let lim x f ðxÞ¼ A. Then
x!1
ð
1
(i) f ðxÞ dx converges if p > 1and A is finite
a
ð
1
(ii) f ðxÞ dx diverges if p @ 1 and A 6¼ 0(A may be infinite).
a
ð 2 2
1 x dx x 1
2
EXAMPLE 1. converges since lim x ¼ .
4
4
0 4x þ 25 x!1 4x þ 25 4
ð
1 xdx x
EXAMPLE 2. p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi diverges since lim x p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1.
4
4
2
2
0 x þ x þ 1 x!1 x þ x þ 1
Similar test can be devised using gðxÞ¼ e tx .
ð
1
3. Series test for integrals with non-negative integrands. f ðxÞ dx converges or diverges accord-
ing as u n , where u n ¼ f ðnÞ,converges or diverges. a
ð ð
1 1
4. Absolute and conditional convergence. f ðxÞ dx is called absolutely convergent if j f ðxÞj dx
a
ð a ð ð
1 1 1
converges. If f ðxÞ dx converges but j f ðxÞj dx diverges, then f ðxÞ dx is called con-
a a a
ditionally convergent.
ð ð
1 1
Theorem 2. If j f ðxÞj dx converges, then f ðxÞ dx converges. In words, an absolutely convergent
a a
integral converges.
ð
1 cos x
EXAMPLE 1. dx is absolutely convergent and thus convergent since
2
a x þ 1
ð ð ð
1 1 dx 1 dx
cos x
dx @ and converges.
2
2
2
0 x þ 1 0 x þ 1 0 x þ 1

ð
ð
1 sin x 1
EXAMPLE 2. dx converges (see Problem 12.11), but sin x dx does not converge (see
0 x 0 x
ð
1 sin x
Problem 12.12). Thus, dx is conditionally convergent.
0 x
Any of the tests used for integrals with non-negative integrands can be used to test for absolute
convergence.