n -f(x) (c) (x - c)*. Ph(x) = Σ k! k=0 Show that for any other polynomial Qn Pn of order n it holds f(x) - Pn(x) f(x) - Qn(x) lim x →C = 0. Interpretation: This shows that the Taylor polynomial of order n is the best possible approximation of f via a polynomial of order n.
n -f(x) (c) (x - c)*. Ph(x) = Σ k! k=0 Show that for any other polynomial Qn Pn of order n it holds f(x) - Pn(x) f(x) - Qn(x) lim x →C = 0. Interpretation: This shows that the Taylor polynomial of order n is the best possible approximation of f via a polynomial of order n.
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
100%
Consider three real numbers a < c < b and a function f : (a, b) → R such that the (n + 1)-th derivative f(n+1) exists on (a,b) and is continuous. Let Pn denote the Taylor polynomial of order n of f around c i.e.
Would you mind showing me the detail, please. Thank you so much
![P₁ (x) = f(k) (c)
k!
Show that for any other polynomial Qn
n
lim
x →C
k=0
·(x – c)k.
Pn of order n it holds
f(x) – Pn(x)
f(x) - Qn(x)
=
0.
Interpretation: This shows that the Taylor polynomial of order n is the best possible approximation of f
via a polynomial of order n.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc30102f3-a996-44b5-a478-64ed6db32292%2Fde8baaeb-aa09-4582-b0c1-f6007c6ce75b%2Fjkb2bhr_processed.png&w=3840&q=75)
Transcribed Image Text:P₁ (x) = f(k) (c)
k!
Show that for any other polynomial Qn
n
lim
x →C
k=0
·(x – c)k.
Pn of order n it holds
f(x) – Pn(x)
f(x) - Qn(x)
=
0.
Interpretation: This shows that the Taylor polynomial of order n is the best possible approximation of f
via a polynomial of order n.
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 3 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)