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Problem 8.1.4. Prove Theorem 8.1.1. Theorem &.1.I. Taylor's serles ythere exats o real number B such thar |f(n+1) (t)| < Bwa t. nonnegative integers n and for on C on an nterval containing a ans X.then lim f(n+1) (t)(x – t)" dt) = 0 n00 n! =a and so fm (a), f(x) = n=0 (x - a)". n!

Problem 8.1.4. Prove Theorem 8.1.1. Theorem &.1.I. Taylor's serles ythere exats o real number B such thar |f(n+1) (t)| < Bwa t. nonnegative integers n and for on C on an nterval containing a ans X.then lim f(n+1) (t)(x – t)" dt) = 0 n00 n! =a and so fm (a), f(x) = n=0 (x - a)". n!

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section: Chapter Questions
Problem 63RE
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Problem 8.1.4. Prove Theorem 8.1.1. Theorem 8.1.1. Taylor's serles ythere exists a real number B such that |f(n+1) (t)| < Bwarnovegine megen n for ali nonnegative integers Tn and for on l on an intervol containing a and C. then f(n+1) (t)(x – – t)" dt ) = 0 lim n00 n! and so f(m) (a) f(x) -(x – a)". n! | n=0 in-context Hint. You might want to use Problem 7.4.8 of Chapter 7. Also there are two cases to consider: a < x and x < a (the case x = a is trivial). You will find that this is true in general. This is why we will often indicate that t is between a and x as in the theorem. In the case x < a. notice that a f(n+1) (t)(x – t)" dt t=Da t%3Dx =|/ fln+1)(t)(t – æ)" dt
Transcribed Image Text:Problem 8.1.4. Prove Theorem 8.1.1. Theorem 8.1.1. Taylor's serles ythere exists a real number B such that |f(n+1) (t)| < Bwarnovegine megen n for ali nonnegative integers Tn and for on l on an intervol containing a and C. then f(n+1) (t)(x – – t)" dt ) = 0 lim n00 n! and so f(m) (a) f(x) -(x – a)". n! | n=0 in-context Hint. You might want to use Problem 7.4.8 of Chapter 7. Also there are two cases to consider: a < x and x < a (the case x = a is trivial). You will find that this is true in general. This is why we will often indicate that t is between a and x as in the theorem. In the case x < a. notice that a f(n+1) (t)(x – t)" dt t=Da t%3Dx =|/ fln+1)(t)(t – æ)" dt
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