[10] Let I be an open interval containing [a, b] and f € Cn+1 (I,R) (ƒ has n+1 continuous derivatives on I). Show that the Taylor formula can be put into the following form: (x – a)* f (x) = E f(k) (a) k! - + Ra (x) k=0 where 1 R, (x) = / (x – t)" f(n+1) (t) dt. n! a Rn (x) is called the integral form of the remainder term. Hint: Use induction, FTC, and integration by parts.
[10] Let I be an open interval containing [a, b] and f € Cn+1 (I,R) (ƒ has n+1 continuous derivatives on I). Show that the Taylor formula can be put into the following form: (x – a)* f (x) = E f(k) (a) k! - + Ra (x) k=0 where 1 R, (x) = / (x – t)" f(n+1) (t) dt. n! a Rn (x) is called the integral form of the remainder term. Hint: Use induction, FTC, and integration by parts.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![(3) [10] Let I be an open interval containing [a, b] and f e C"+1 (I,R) (ƒ has n+1 continuous
derivatives on I). Show that the Taylor formula can be put into the following form:
n
f (x) = f(k) (a)
k!
(x – a)*
+ Rn (x)
k=0
where
Rn (x)
/ (x – t)" f(a+1) (t) dt.
a
Rn (x) is called the integral form of the remainder term. Hint: Use induction, FTC, and
integration by parts.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc8e6d565-5f5b-4283-af37-3b3705eb4576%2F1e9ba0f6-75ab-4ce0-8ac5-83f1a81fd060%2F1xaxgh_processed.png&w=3840&q=75)
Transcribed Image Text:(3) [10] Let I be an open interval containing [a, b] and f e C"+1 (I,R) (ƒ has n+1 continuous
derivatives on I). Show that the Taylor formula can be put into the following form:
n
f (x) = f(k) (a)
k!
(x – a)*
+ Rn (x)
k=0
where
Rn (x)
/ (x – t)" f(a+1) (t) dt.
a
Rn (x) is called the integral form of the remainder term. Hint: Use induction, FTC, and
integration by parts.
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