Proposition 7.2. Interchanging integral and limit Suppose fn E R[a, b] and fn → f uniformly on [a, b]. Then f E R[a, b] and . fn → S, f.
Proposition 7.2. Interchanging integral and limit Suppose fn E R[a, b] and fn → f uniformly on [a, b]. Then f E R[a, b] and . fn → S, f.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Proposition 7.2. Interchanging integral and limit
Suppose fn E R[a, b] and fn → f uniformly on [a, b].
Then f E R[a, b] and fn → S" f.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9b07bcd4-aafe-40be-8b09-28ec967f442e%2F0e75c917-4e26-498d-90b0-909c8fd2b010%2Fc98diis_processed.png&w=3840&q=75)
Transcribed Image Text:Proposition 7.2. Interchanging integral and limit
Suppose fn E R[a, b] and fn → f uniformly on [a, b].
Then f E R[a, b] and fn → S" f.
![7. Prove the following theorem (which is a version of Proposition 7.2 valid for the infinite interval
[a, 0). )
Suppose that (fn); f, g are regulated functions on [0, R] for every R > 0 and satisfy:
(i) fn → f uniformly on [0, R], (ii) |fn(x)| < g(x) on [0, ∞), and (iii) ſº g is convergent.
Show that the improper integral f f exists and fn →
f as n → o.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9b07bcd4-aafe-40be-8b09-28ec967f442e%2F0e75c917-4e26-498d-90b0-909c8fd2b010%2Fxcqekkf_processed.png&w=3840&q=75)
Transcribed Image Text:7. Prove the following theorem (which is a version of Proposition 7.2 valid for the infinite interval
[a, 0). )
Suppose that (fn); f, g are regulated functions on [0, R] for every R > 0 and satisfy:
(i) fn → f uniformly on [0, R], (ii) |fn(x)| < g(x) on [0, ∞), and (iii) ſº g is convergent.
Show that the improper integral f f exists and fn →
f as n → o.
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 2 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
![Introductory Mathematics for Engineering Applicat…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
![Introductory Mathematics for Engineering Applicat…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Mathematics For Machine Technology](https://www.bartleby.com/isbn_cover_images/9781337798310/9781337798310_smallCoverImage.jpg)
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
![Basic Technical Mathematics](https://www.bartleby.com/isbn_cover_images/9780134437705/9780134437705_smallCoverImage.gif)
![Topology](https://www.bartleby.com/isbn_cover_images/9780134689517/9780134689517_smallCoverImage.gif)