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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_09” 2011/5/22 23:21 page 13 #13
Sequences and Series of Functions 9-13
If |s
n
(x)|≤M, then
|t
n
(x) t
m
(x)|≤M
n
k=m+1
(g
k
(x) g
k+1
(x)) + Mg
n+1
(x) + Mg
m+1
(x)
= M(g
m+1
(x) g
n+1
(x)) + Mg
n+1
(x) + Mg
m+1
(x)
= 2Mg
m+1
(x). (9.2)
Since g
n
0asn →∞uniformly on E, (9.2) shows that {t
n
} is uni-
formly Cauchy on E and by Theorem 9.3.7, it is uniformly convergent
on E.
Here is an example of a series that can be proved to be uniformly
convergent using the above test but not by using Weierstrass M-test.
Example 9.3.20 Consider
s
n
(x) =
n
k=1
e
ikx
=
e
ix
(1 e
inx
)
1 e
ix
=
e
ix
e
inx/2
e
inx/2
e
inx/2
e
ix/2
e
ix/2
e
ix/2
=
sin
nx
2
sin
x
2
e
i(n+1)x/2
.
It now follows
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Publisher Resources

ISBN: 9781299447561Publisher Website