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Problem 8.1.3. Prove Lemma 8.1.2. Lemma 8.1.2. / and are integroble functions and a < b.mem then | f(t) dt < |f(t)| dt. t=a t=a in-context

Problem 8.1.3. Prove Lemma 8.1.2. Lemma 8.1.2. / and are integroble functions and a < b.mem then | f(t) dt < |f(t)| dt. t=a t=a in-context

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.3: The Addition And Subtraction Formulas
Problem 79E
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Real math analysis, I need help with problem 8.1.3 please.

Theorem 8.1.1. Taylor's series If there exists a real number B such that f(n+1) (t)| < B for all nonnegative integers n and for allt on an interval containing a and x then 1 lim G fln+ (t)(x – t)" dt) = 0 n! t=a and so Σ f(m) (a) f(x) = a)". %3D n! n=0 In order to prove this, it might help to first prove the following. Lemma 8.1.2. If f and f are integrable functions and a < b then f(t) dt < |f(t)| dt. t=a %3a Problem 8.1.3. Prove Lemma 8.1.2. Lemma 8.1.2. / and IJ are integrable punctions and a < bmem f(t) dt </ |F(t)| dt. t=a t=a in-contet Hint. L-|f(t)| < f(t) < |f(t)|
Transcribed Image Text:Theorem 8.1.1. Taylor's series If there exists a real number B such that f(n+1) (t)| < B for all nonnegative integers n and for allt on an interval containing a and x then 1 lim G fln+ (t)(x – t)" dt) = 0 n! t=a and so Σ f(m) (a) f(x) = a)". %3D n! n=0 In order to prove this, it might help to first prove the following. Lemma 8.1.2. If f and f are integrable functions and a < b then f(t) dt < |f(t)| dt. t=a %3a Problem 8.1.3. Prove Lemma 8.1.2. Lemma 8.1.2. / and IJ are integrable punctions and a < bmem f(t) dt </ |F(t)| dt. t=a t=a in-contet Hint. L-|f(t)| < f(t) < |f(t)|
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