Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
(b)Please solve the problem
![Solutions
(a) We apply the Integral Test and find that
np
u = x, du = 2x dx
xp
= lim
n_
boAG
1.
2e
1.
2e
= lim
2eb
Since the integral converges, the series also converges.
(b) Again applying the Integral Test,
xp
2 In x
np nə
u = In x, x = e", dx = e" du
%3D
%3D
%3D
uj 7
%3D
1.
lim
(e/2) > 1
-
%3D
%3D
b→08 In (특)
1.
The improper integral diverges, so the series diverges also.
Error Estimation
For some convergent series, such as the geometric series or the t
Example 5 of Section 9.2, we can actually find the total sum of th
can find the limiting value S of the sequence of partial sums. For mo
however, we cannot easily find the total sum. Nevertheless, we can
adding the first n terms to get sn, but we need to know how far of
sum S. An approximation to a function or to a number is more uset
panied by a bound on the size of the worst possible error that could
error bound we can try to make an estimate or approximation tha](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7a74984e-ce90-4333-bb94-ec5cb474eff3%2F7f4aa45b-20ad-46cd-b4cb-0cf34f3ab919%2Fqa7yd0y_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Solutions
(a) We apply the Integral Test and find that
np
u = x, du = 2x dx
xp
= lim
n_
boAG
1.
2e
1.
2e
= lim
2eb
Since the integral converges, the series also converges.
(b) Again applying the Integral Test,
xp
2 In x
np nə
u = In x, x = e", dx = e" du
%3D
%3D
%3D
uj 7
%3D
1.
lim
(e/2) > 1
-
%3D
%3D
b→08 In (특)
1.
The improper integral diverges, so the series diverges also.
Error Estimation
For some convergent series, such as the geometric series or the t
Example 5 of Section 9.2, we can actually find the total sum of th
can find the limiting value S of the sequence of partial sums. For mo
however, we cannot easily find the total sum. Nevertheless, we can
adding the first n terms to get sn, but we need to know how far of
sum S. An approximation to a function or to a number is more uset
panied by a bound on the size of the worst possible error that could
error bound we can try to make an estimate or approximation tha
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