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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_03” 2011/5/22 22:50 page 31 #31
Sequences and Series 3-31
We now deduce the following Corollary from Theorem 3.4.7.
Corollary 3.4.10 (Cauchy’s Theorem on product of series) If
n=0
a
n
and
n=0
b
n
both converge absolutely and if
n=0
c
n
is the Cauchy product
of
n=0
a
n
and
n=0
b
n
, then
n=0
c
n
converges absolutely and we have
n=0
c
n
=
n=0
a
n

n=0
b
n
.
Proof Since both the series
n=0
a
n
and
n=0
b
n
converge absolutely we
have that
n=0
c
n
converges and the required equality holds (by using
Merten’s Theorem). Hence it is enough to show that
n=0
c
n
converges
absolutely. Let d
n
=
n
k=0
|a
k
||b
nk
|so that
n=0
d
n
is the Cauchy product
of
n=0
|a
n
| and
n=0
|b
n
|. Then again by Merten’s Theorem, the series
n=0
d
n
converges (note that a positive-termed series ...
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Publisher Resources

ISBN: 9781299447561Publisher Website