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Real Analysis
book

Real Analysis

by V. Karunakaran
May 2024
Intermediate to advanced content levelIntermediate to advanced
585 pages
15h 38m
English
Pearson India
Content preview from Real Analysis
“real: chapter_08” 2011/5/22 23:34 page 23 #23
Riemann Integration 8-23
Similarly using φ : [−M , M ]→R, given by φ(x) =|x|,wesee
that φ f =|f |∈
R[a, b]. This proves (i) and the first part of
(ii). To complete the proof, we observe that f (x) ≤|f (x)| and
f (x) ≤|f (x)| together imply (use Theorem 8.5.2) that
b
a
f (x)dx
b
a
|f (x)|dx and
b
a
f (x)dx
b
a
|f (x)|dx. It now follows that
b
a
|f (x)|dx
b
a
f (x) dx
b
a
|f (x)|dx
which is equivalent to
b
a
f (x)dx
b
a
|f (x)|dx.
Note 8.5.7 We now recast Taylor’s theorem obtained in Chapter 6 by
expressing the reminder term as an integral.
Theorem 8.5.8 (Taylor’s theorem with integral form ...
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Publisher Resources

ISBN: 9781299447561Publisher Website